So I'm thinking about Factorial/Exponential Identity,
from 2003, and one of the outputs is like this.

n! ~= root( sum n^n - (sum n) ^n )

Now this amuses me a lot, because, it relates these
different terms, or rather, in terms of one variable
a term, by these various operations, relating the
operations together.

Here n! is n factorial and root is the square root
and sum n^n is sum_i=1^n i^i and (sum n)^n is
(n(n+1)/2)^n.

So it's funny because as of a root of a difference,
it's sort of like a mean

arithmetic mean = n'th-part sum-n-terms
geometric mean = n'th-root product-n-terms

but just about "roots of difference, of various terms".

Then, ~= means "approximately equal" it's an approximation,
or "it's true in the limit", here leaving that out leaves
these kinds of things.

F = n!
L = sum (n^n)
R = (sum n)^n = (n(n+1)/2)^n

The most simple sort of algebraic manipulation
leads for these kinds of things.

F^2 = L - R

L = F^2 + R

R = L - F^2

2^n (L - F^2) = (n(n+1))^n

n'th-root (2^n (L - F^2)) = n^2 + n

n'th-root (2^n (L - F^2)) = X

n^2 + n - X = 0

Now I'm used to seeing something like

(n + 1/2)^2 = n^2 + n + 1/4, it's a square

which gets me to wondering that

X = -1/4

when, "n is a root of this quadratic".

n'th-root (2^n (L - F^2)) = -1/4

(-1/4)^n = 2^n (L - F^2)

(-1/8)^n = L - F^2

(-1/8)^n = sum (n^n) - n!^2

Borel versus Combinatorics, anyone?

Fun with Euler? In the limit, ....

Yet, for keeping things real, is for something like

n^2 + n - X = (n+b)(n-a), where b = a + 1

= n^2 + n - ab, or for where

X = ab = a(a+1) = a^2 + a

when, "n is a root of this quadratic".

n'th-root (2^n (L - F^2)) = ab

(ab)^n = 2^n (L - F^2)

(ab/2)^n = L - F^2

(ab/2)^n = sum (n^n) - n!^2

((a^2 + a)/2)^n = sum (n^n) - n!^2

Borel versus Combinatorics, anyone?

Fun with Euler? In the limit, ....

((a^2) + a)/2) = A

A^n = L - F^2

A^n/L = 1 - F^2/L

Hmm...

So, looking for a(a+1) = 1,

a^2 + a - 1 = 0

I lame out and turn to root finder and inverse symbolic calculator.

(for b^2 -4ac/2a, ....)

a = (root 5 - 1)/2 ~ 0.618
a = (-root5 -1)/2 ~= -1.618

https://en.wikipedia.org/wiki/Golden_ratio

Hmm, you know they say phi the Golden Mean shows up
everywhere, but, the "negative Golden Mean"?

Then this, "Golden mean minus one"?

So, calling those -phi and phi-1,

-phi:

n^2 + n + (-phi)(-phi+1) = 0

n^2 +n + phi^2 - phi = 0

phi-1:

n^2 + n + phi^2 = 0

X = phi^2, when, "n is a root of this quadratic".

Keeping things positive,

phi-1:

phi^2 = X
phi^2 = n'th-root (2^n (L - F^2))

Great, now it looks like I added the Golden Mean among
the "Factorial/Exponential Identities" given some various
ideas in laws of large numbers, a square compact Cantor space,
some expectations in probability, and some sorts these other things.

Borel vs. combinatorics, anyone?

On 2/16/2024 8:30 PM, Ross Finlayson wrote:
>
>
> So I'm thinking about Factorial/Exponential Identity,
> from 2003, and one of the outputs is like this.
>
> n! ~=  root(  sum n^n  - (sum n) ^n )
>
> Now this amuses me a lot, because, it relates these
> different terms, or rather, in terms of one variable
> a term, by these various operations, relating the
> operations together.
>
> Here n! is n factorial and root is the square root
> and sum n^n is sum_i=1^n i^i and (sum n)^n is
> (n(n+1)/2)^n.
[...]

Fun with roots... Storing actual data in them:

https://groups.google.com/g/comp.lang.c++/c/bB1wA4wvoFc/m/GdzmMd41AQAJ

;^)

On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>
>
> So I'm thinking about Factorial/Exponential Identity,
> from 2003, and one of the outputs is like this.
>
> n! ~= root( sum n^n - (sum n) ^n )
>
> Now this amuses me a lot, because, it relates these
> different terms, or rather, in terms of one variable
> a term, by these various operations, relating the
> operations together.
>
> Here n! is n factorial and root is the square root
> and sum n^n is sum_i=1^n i^i and (sum n)^n is
> (n(n+1)/2)^n.
>
> So it's funny because as of a root of a difference,
> it's sort of like a mean
>
> arithmetic mean = n'th-part sum-n-terms
> geometric mean = n'th-root product-n-terms
>
> but just about "roots of difference, of various terms".
>
> Then, ~= means "approximately equal" it's an approximation,
> or "it's true in the limit", here leaving that out leaves
> these kinds of things.
>
> F = n!
> L = sum (n^n)
> R = (sum n)^n = (n(n+1)/2)^n
>
> The most simple sort of algebraic manipulation
> leads for these kinds of things.
>
> F^2 = L - R
>
> L = F^2 + R
>
> R = L - F^2
>
> 2^n (L - F^2) = (n(n+1))^n
>
> n'th-root (2^n (L - F^2)) = n^2 + n
>
> n'th-root (2^n (L - F^2)) = X
>
>
> n^2 + n - X = 0
>
> Now I'm used to seeing something like
>
> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>
> which gets me to wondering that
>
> X = -1/4
>
> when, "n is a root of this quadratic".
>
> n'th-root (2^n (L - F^2)) = -1/4
>
> (-1/4)^n = 2^n (L - F^2)
>
> (-1/8)^n = L - F^2
>
> (-1/8)^n = sum (n^n) - n!^2
>
> Borel versus Combinatorics, anyone?
>
> Fun with Euler? In the limit, ....
>
> Yet, for keeping things real, is for something like
>
> n^2 + n - X = (n+b)(n-a), where b = a + 1
>
> = n^2 + n - ab, or for where
>
> X = ab = a(a+1) = a^2 + a
>
> when, "n is a root of this quadratic".
>
> n'th-root (2^n (L - F^2)) = ab
>
> (ab)^n = 2^n (L - F^2)
>
> (ab/2)^n = L - F^2
>
> (ab/2)^n = sum (n^n) - n!^2
>
> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>
> Borel versus Combinatorics, anyone?
>
> Fun with Euler? In the limit, ....
>
> ((a^2) + a)/2) = A
>
> A^n = L - F^2
>
> A^n/L = 1 - F^2/L
>
> Hmm...
>
> So, looking for a(a+1) = 1,
>
> a^2 + a - 1 = 0
>
> I lame out and turn to root finder and inverse symbolic calculator.
>
> (for b^2 -4ac/2a, ....)
>
> a = (root 5 - 1)/2 ~ 0.618
> a = (-root5 -1)/2 ~= -1.618
>
>
>
> https://en.wikipedia.org/wiki/Golden_ratio
>
> Hmm, you know they say phi the Golden Mean shows up
> everywhere, but, the "negative Golden Mean"?
>
> Then this, "Golden mean minus one"?
>
> So, calling those -phi and phi-1,
>
> -phi:
>
> n^2 + n + (-phi)(-phi+1) = 0
>
> n^2 +n + phi^2 - phi = 0
>
> phi-1:
>
> n^2 + n + phi^2 = 0
>
> X = phi^2, when, "n is a root of this quadratic".
>
> Keeping things positive,
>
> phi-1:
>
> phi^2 = X
> phi^2 = n'th-root (2^n (L - F^2))
>
> Great, now it looks like I added the Golden Mean among
> the "Factorial/Exponential Identities" given some various
> ideas in laws of large numbers, a square compact Cantor space,
> some expectations in probability, and some sorts these other things.
>
> Borel vs. combinatorics, anyone?

Well, whuh, the phi-1 case, ...

n^2 + n + (phi-1)(phi) = 0

looks the same as the -phi case, about that,

-phi:
n^2 + n + (-phi)(-phi+1) = 0
n^2 + n + phi^2 - phi = 0

phi-1:
n^2 + n + (phi-1)(phi) = 0
n^2 + n + phi^2 - phi = 0

It looks in both cases that

phi^2 - phi = X

X = n'th-root (2^n (L - F^2))

lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi

lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2

Because, phi-1 = 1/phi,

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2

1/phi:

n^2 + n + (1/phi)(1/phi+1) = 0
n^2 + n + (1/phi)(1/phi+phi/phi) = 0
n^2 + n + (1/phi)(1/phi+phi/phi) = 0
n^2 + n + (1/phi^2 + phi/phi^2) = 0
n^2 + n + ((1 +phi)/phi^2) = 0

(1+phi)/phi^2 = 1/phi^2 + 1/phi

X = phi^2, when, "n is a root of this quadratic".

-1 -1/phi:

n^2 + n + (-1 -1/phi)(-1/phi) = 0
n^2 + n + 1/phi + 1/phi^2 = 0
1/phi + 1/phi^2 = 1

Hmm, phi: came and went, yielded:

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2

So now I'm looking at the Lucas number,
or that "phi = 1 + 1/phi can be expanded
recursively to obtain a continued fraction
for the golden ratio".

This is where basically is for getting out
the "infinity'th root" of phi, then to put it in.

"... one can easily decompose any power of phi into
a multiple of phi and a constant. The multiple
and the constant are always adjacent Fibonacci numbers."

So I'm looking for the "negative Lucas and Fibonacci
numbers", that phi^n has an expression in L_n and F_n,
looking for phi^{-n}.

"That pi times root phi is _close_ to 4 is only called
'numerical coincidence'."

Let's see, if root phi is "the Golden Root", then,
looking for powers of the Golden Root, ....

Hmm, phi: came and went, yielded:

lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2

Yet, any sort expression n'th powers of phi goes in, ....

X = phi^2 - phi

phi^2 - phi - X = 0

On 02/16/2024 09:22 PM, Ross Finlayson wrote:
> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>
>>
>> So I'm thinking about Factorial/Exponential Identity,
>> from 2003, and one of the outputs is like this.
>>
>> n! ~= root( sum n^n - (sum n) ^n )
>>
>> Now this amuses me a lot, because, it relates these
>> different terms, or rather, in terms of one variable
>> a term, by these various operations, relating the
>> operations together.
>>
>> Here n! is n factorial and root is the square root
>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>> (n(n+1)/2)^n.
>>
>> So it's funny because as of a root of a difference,
>> it's sort of like a mean
>>
>> arithmetic mean = n'th-part sum-n-terms
>> geometric mean = n'th-root product-n-terms
>>
>> but just about "roots of difference, of various terms".
>>
>> Then, ~= means "approximately equal" it's an approximation,
>> or "it's true in the limit", here leaving that out leaves
>> these kinds of things.
>>
>> F = n!
>> L = sum (n^n)
>> R = (sum n)^n = (n(n+1)/2)^n
>>
>> The most simple sort of algebraic manipulation
>> leads for these kinds of things.
>>
>> F^2 = L - R
>>
>> L = F^2 + R
>>
>> R = L - F^2
>>
>> 2^n (L - F^2) = (n(n+1))^n
>>
>> n'th-root (2^n (L - F^2)) = n^2 + n
>>
>> n'th-root (2^n (L - F^2)) = X
>>
>>
>> n^2 + n - X = 0
>>
>> Now I'm used to seeing something like
>>
>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>
>> which gets me to wondering that
>>
>> X = -1/4
>>
>> when, "n is a root of this quadratic".
>>
>> n'th-root (2^n (L - F^2)) = -1/4
>>
>> (-1/4)^n = 2^n (L - F^2)
>>
>> (-1/8)^n = L - F^2
>>
>> (-1/8)^n = sum (n^n) - n!^2
>>
>> Borel versus Combinatorics, anyone?
>>
>> Fun with Euler? In the limit, ....
>>
>> Yet, for keeping things real, is for something like
>>
>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>
>> = n^2 + n - ab, or for where
>>
>> X = ab = a(a+1) = a^2 + a
>>
>> when, "n is a root of this quadratic".
>>
>> n'th-root (2^n (L - F^2)) = ab
>>
>> (ab)^n = 2^n (L - F^2)
>>
>> (ab/2)^n = L - F^2
>>
>> (ab/2)^n = sum (n^n) - n!^2
>>
>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>
>> Borel versus Combinatorics, anyone?
>>
>> Fun with Euler? In the limit, ....
>>
>> ((a^2) + a)/2) = A
>>
>> A^n = L - F^2
>>
>> A^n/L = 1 - F^2/L
>>
>> Hmm...
>>
>> So, looking for a(a+1) = 1,
>>
>> a^2 + a - 1 = 0
>>
>> I lame out and turn to root finder and inverse symbolic calculator.
>>
>> (for b^2 -4ac/2a, ....)
>>
>> a = (root 5 - 1)/2 ~ 0.618
>> a = (-root5 -1)/2 ~= -1.618
>>
>>
>>
>> https://en.wikipedia.org/wiki/Golden_ratio
>>
>> Hmm, you know they say phi the Golden Mean shows up
>> everywhere, but, the "negative Golden Mean"?
>>
>> Then this, "Golden mean minus one"?
>>
>> So, calling those -phi and phi-1,
>>
>> -phi:
>>
>> n^2 + n + (-phi)(-phi+1) = 0
>>
>> n^2 +n + phi^2 - phi = 0
>>
>> phi-1:
>>
>> n^2 + n + phi^2 = 0
>>
>> X = phi^2, when, "n is a root of this quadratic".
>>
>> Keeping things positive,
>>
>> phi-1:
>>
>> phi^2 = X
>> phi^2 = n'th-root (2^n (L - F^2))
>>
>> Great, now it looks like I added the Golden Mean among
>> the "Factorial/Exponential Identities" given some various
>> ideas in laws of large numbers, a square compact Cantor space,
>> some expectations in probability, and some sorts these other things.
>>
>> Borel vs. combinatorics, anyone?
>
>
>
>
> Well, whuh, the phi-1 case, ...
>
> n^2 + n + (phi-1)(phi) = 0
>
> looks the same as the -phi case, about that,
>
> -phi:
> n^2 + n + (-phi)(-phi+1) = 0
> n^2 + n + phi^2 - phi = 0
>
> phi-1:
> n^2 + n + (phi-1)(phi) = 0
> n^2 + n + phi^2 - phi = 0
>
> It looks in both cases that
>
> phi^2 - phi = X
>
> X = n'th-root (2^n (L - F^2))
>
> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>
> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>
> Because, phi-1 = 1/phi,
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>
>
> 1/phi:
>
> n^2 + n + (1/phi)(1/phi+1) = 0
> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
> n^2 + n + (1/phi^2 + phi/phi^2) = 0
> n^2 + n + ((1 +phi)/phi^2) = 0
>
> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>
> X = phi^2, when, "n is a root of this quadratic".
>
> -1 -1/phi:
>
> n^2 + n + (-1 -1/phi)(-1/phi) = 0
> n^2 + n + 1/phi + 1/phi^2 = 0
> 1/phi + 1/phi^2 = 1
>
>
> Hmm, phi: came and went, yielded:
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>
> So now I'm looking at the Lucas number,
> or that "phi = 1 + 1/phi can be expanded
> recursively to obtain a continued fraction
> for the golden ratio".
>
> This is where basically is for getting out
> the "infinity'th root" of phi, then to put it in.
>
> "... one can easily decompose any power of phi into
> a multiple of phi and a constant. The multiple
> and the constant are always adjacent Fibonacci numbers."
>
> So I'm looking for the "negative Lucas and Fibonacci
> numbers", that phi^n has an expression in L_n and F_n,
> looking for phi^{-n}.
>
> "That pi times root phi is _close_ to 4 is only called
> 'numerical coincidence'."
>
> Let's see, if root phi is "the Golden Root", then,
> looking for powers of the Golden Root, ....
>
>
> Hmm, phi: came and went, yielded:
>
> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>
> Yet, any sort expression n'th powers of phi goes in, ....
>
> X = phi^2 - phi
>
> phi^2 - phi - X = 0
>
>
>
>

The Wiki for phi says an interesting thing, about
the consistently small terms in its contined fraction,
as explain why the approximates to phi^{-1} converge
so slowly: that "this makes the golden ratio an extreme
case of the Hurwitz inequality for Diophantine approximations."

So looking around some guy says Binet's formulas are like so:

Lucas number L_n = phi^n + (-phi)^{-n}

Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}

Then, I'm looking for phi^{-n}, and n goes to infinity,
basically for what are these "negative Lucas and Fibonacci
numbers", L_-n and F_-n, and what are these "inverse Lucas
and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
looking for phi^{1/n}.

"The ratios between consecutive Fibonacci numbers
approaches phi...", in the limit as n -> \infty.

I suppose the roots in binary are the same as
the roots in "phi-nary". I.e., in their representations.

"A root-phi rectangle divides into a pair
of Kepler triangles (right triangles with
edge lengths in geometric progression)."
-- https://en.wikipedia.org/wiki/Dynamic_rectangle

https://en.wikipedia.org/wiki/Kepler_triangle

1^2 + root-phi ^2 = phi^2

root phi = root (phi^2 - 1)

"The three edge length 1, root phi, and phi are
the harmonic mean, geometric mean, and arithmetic mean,
respectively, of the two numbers phi +- 1."

https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/

About 25/4 and 5/2, ....

https://mathworld.wolfram.com/GoldenRatio.html

phi = 2 cos (pi/5)

Not much to be found about "roots of phi".
Or, there's tons to be found "roots of phi" historically,
not so much analytically.

On 02/17/2024 09:24 AM, Ross Finlayson wrote:
> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>
>>>
>>> So I'm thinking about Factorial/Exponential Identity,
>>> from 2003, and one of the outputs is like this.
>>>
>>> n! ~= root( sum n^n - (sum n) ^n )
>>>
>>> Now this amuses me a lot, because, it relates these
>>> different terms, or rather, in terms of one variable
>>> a term, by these various operations, relating the
>>> operations together.
>>>
>>> Here n! is n factorial and root is the square root
>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>> (n(n+1)/2)^n.
>>>
>>> So it's funny because as of a root of a difference,
>>> it's sort of like a mean
>>>
>>> arithmetic mean = n'th-part sum-n-terms
>>> geometric mean = n'th-root product-n-terms
>>>
>>> but just about "roots of difference, of various terms".
>>>
>>> Then, ~= means "approximately equal" it's an approximation,
>>> or "it's true in the limit", here leaving that out leaves
>>> these kinds of things.
>>>
>>> F = n!
>>> L = sum (n^n)
>>> R = (sum n)^n = (n(n+1)/2)^n
>>>
>>> The most simple sort of algebraic manipulation
>>> leads for these kinds of things.
>>>
>>> F^2 = L - R
>>>
>>> L = F^2 + R
>>>
>>> R = L - F^2
>>>
>>> 2^n (L - F^2) = (n(n+1))^n
>>>
>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>
>>> n'th-root (2^n (L - F^2)) = X
>>>
>>>
>>> n^2 + n - X = 0
>>>
>>> Now I'm used to seeing something like
>>>
>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>
>>> which gets me to wondering that
>>>
>>> X = -1/4
>>>
>>> when, "n is a root of this quadratic".
>>>
>>> n'th-root (2^n (L - F^2)) = -1/4
>>>
>>> (-1/4)^n = 2^n (L - F^2)
>>>
>>> (-1/8)^n = L - F^2
>>>
>>> (-1/8)^n = sum (n^n) - n!^2
>>>
>>> Borel versus Combinatorics, anyone?
>>>
>>> Fun with Euler? In the limit, ....
>>>
>>> Yet, for keeping things real, is for something like
>>>
>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>
>>> = n^2 + n - ab, or for where
>>>
>>> X = ab = a(a+1) = a^2 + a
>>>
>>> when, "n is a root of this quadratic".
>>>
>>> n'th-root (2^n (L - F^2)) = ab
>>>
>>> (ab)^n = 2^n (L - F^2)
>>>
>>> (ab/2)^n = L - F^2
>>>
>>> (ab/2)^n = sum (n^n) - n!^2
>>>
>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>
>>> Borel versus Combinatorics, anyone?
>>>
>>> Fun with Euler? In the limit, ....
>>>
>>> ((a^2) + a)/2) = A
>>>
>>> A^n = L - F^2
>>>
>>> A^n/L = 1 - F^2/L
>>>
>>> Hmm...
>>>
>>> So, looking for a(a+1) = 1,
>>>
>>> a^2 + a - 1 = 0
>>>
>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>
>>> (for b^2 -4ac/2a, ....)
>>>
>>> a = (root 5 - 1)/2 ~ 0.618
>>> a = (-root5 -1)/2 ~= -1.618
>>>
>>>
>>>
>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>
>>> Hmm, you know they say phi the Golden Mean shows up
>>> everywhere, but, the "negative Golden Mean"?
>>>
>>> Then this, "Golden mean minus one"?
>>>
>>> So, calling those -phi and phi-1,
>>>
>>> -phi:
>>>
>>> n^2 + n + (-phi)(-phi+1) = 0
>>>
>>> n^2 +n + phi^2 - phi = 0
>>>
>>> phi-1:
>>>
>>> n^2 + n + phi^2 = 0
>>>
>>> X = phi^2, when, "n is a root of this quadratic".
>>>
>>> Keeping things positive,
>>>
>>> phi-1:
>>>
>>> phi^2 = X
>>> phi^2 = n'th-root (2^n (L - F^2))
>>>
>>> Great, now it looks like I added the Golden Mean among
>>> the "Factorial/Exponential Identities" given some various
>>> ideas in laws of large numbers, a square compact Cantor space,
>>> some expectations in probability, and some sorts these other things.
>>>
>>> Borel vs. combinatorics, anyone?
>>
>>
>>
>>
>> Well, whuh, the phi-1 case, ...
>>
>> n^2 + n + (phi-1)(phi) = 0
>>
>> looks the same as the -phi case, about that,
>>
>> -phi:
>> n^2 + n + (-phi)(-phi+1) = 0
>> n^2 + n + phi^2 - phi = 0
>>
>> phi-1:
>> n^2 + n + (phi-1)(phi) = 0
>> n^2 + n + phi^2 - phi = 0
>>
>> It looks in both cases that
>>
>> phi^2 - phi = X
>>
>> X = n'th-root (2^n (L - F^2))
>>
>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>
>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>
>> Because, phi-1 = 1/phi,
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>
>>
>> 1/phi:
>>
>> n^2 + n + (1/phi)(1/phi+1) = 0
>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>> n^2 + n + ((1 +phi)/phi^2) = 0
>>
>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>
>> X = phi^2, when, "n is a root of this quadratic".
>>
>> -1 -1/phi:
>>
>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>> n^2 + n + 1/phi + 1/phi^2 = 0
>> 1/phi + 1/phi^2 = 1
>>
>>
>> Hmm, phi: came and went, yielded:
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>
>> So now I'm looking at the Lucas number,
>> or that "phi = 1 + 1/phi can be expanded
>> recursively to obtain a continued fraction
>> for the golden ratio".
>>
>> This is where basically is for getting out
>> the "infinity'th root" of phi, then to put it in.
>>
>> "... one can easily decompose any power of phi into
>> a multiple of phi and a constant. The multiple
>> and the constant are always adjacent Fibonacci numbers."
>>
>> So I'm looking for the "negative Lucas and Fibonacci
>> numbers", that phi^n has an expression in L_n and F_n,
>> looking for phi^{-n}.
>>
>> "That pi times root phi is _close_ to 4 is only called
>> 'numerical coincidence'."
>>
>> Let's see, if root phi is "the Golden Root", then,
>> looking for powers of the Golden Root, ....
>>
>>
>> Hmm, phi: came and went, yielded:
>>
>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>
>> Yet, any sort expression n'th powers of phi goes in, ....
>>
>> X = phi^2 - phi
>>
>> phi^2 - phi - X = 0
>>
>>
>>
>>
>
>
>
> The Wiki for phi says an interesting thing, about
> the consistently small terms in its contined fraction,
> as explain why the approximates to phi^{-1} converge
> so slowly: that "this makes the golden ratio an extreme
> case of the Hurwitz inequality for Diophantine approximations."
>
>
> So looking around some guy says Binet's formulas are like so:
>
> Lucas number L_n = phi^n + (-phi)^{-n}
>
> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>
> Then, I'm looking for phi^{-n}, and n goes to infinity,
> basically for what are these "negative Lucas and Fibonacci
> numbers", L_-n and F_-n, and what are these "inverse Lucas
> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
> looking for phi^{1/n}.
>
>
>
> "The ratios between consecutive Fibonacci numbers
> approaches phi...", in the limit as n -> \infty.
>
> I suppose the roots in binary are the same as
> the roots in "phi-nary". I.e., in their representations.
>
> "A root-phi rectangle divides into a pair
> of Kepler triangles (right triangles with
> edge lengths in geometric progression)."
> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>
> https://en.wikipedia.org/wiki/Kepler_triangle
>
> 1^2 + root-phi ^2 = phi^2
>
> root phi = root (phi^2 - 1)
>
> "The three edge length 1, root phi, and phi are
> the harmonic mean, geometric mean, and arithmetic mean,
> respectively, of the two numbers phi +- 1."
>
>
> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>
>
> About 25/4 and 5/2, ....
>
> https://mathworld.wolfram.com/GoldenRatio.html
>
> phi = 2 cos (pi/5)
>
> Not much to be found about "roots of phi".
> Or, there's tons to be found "roots of phi" historically,
> not so much analytically.
>
> https://arxiv.org/abs/2312.10767 :
>
> "Interestingly, in the non-rotating limit, RFD bifurcates
> into the usual, non-rotating FD branch and into a spurious
> branch, named "golden" branch, mapping a non-rotating
> (static) extremal BH to an under-rotating (stationary)
> extremal BH, in which the ratio between the angular
> momentum and the non-rotating entropy is the square
> root of the golden ratio."
>
> https://arxiv.org/abs/2210.13395
>
> Found some interest Nigerian papers about
> "inverse powers of phi", phi^{-n}, but looking
> for "roots of phi", phi^{1/n}.
>
>
> https://arxiv.org/abs/1512.00379
>
> "Psi is the square root of the golden ratio."
>
> Lakshmi Roychowdhury, Graf and Luschgy,
> https://link.springer.com/book/10.1007/BFb0103945
> still looking for "n'th roots of phi".
>
> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>
> phi = 2 cos (pi/5)
> phi/2 = cos(pi/5)
>
> nth-root phi/2 = nth-root cos(pi/5)
>
> 1/phi = phi - 1
> phi + 1 - 1 = 2 cos (pi/5)
> 1/phi + 1 = 2 cos (pi/5)
> 1/phi = 2 cos (pi/5) - 1
>
> phi = 1/ ( 2cos (pi/5) - 1
> phi/2 = cos (pi/5)
> arccos (phi/2) = pi/5
> pi = 5 arccos (phi/2)
>
> arccos(-phi/2) = 0
> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>
> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
> sin(pi/5) = root (1- (phi/2)^2 )
>
> cos^2(theta) = cos(pi/5)
> sin(theta) = +- root( 1-cos(pi/5) )
> sin(theta) = += root (1-phi/2)
> theta = +- arcsin(root(1-phi/2))
> cos(theta) = root(phi)
> cos(arcsin(root(1-phi/2))) = root(phi)
>
> root phi = 1.27202....
> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>
>
> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root 5)/2.
>
> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
> Of course that's just modular and doesn't say anything about roots.
>
> What I'm looking at is:
>
> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>
> with the idea to write phi^2 - phi in terms of an infinite series,
> then get all the terms on one side. Then what I figure is that
> it would be in terms of the n'th root or to-the-1-over-n,
> getting those on one side, or just the phi part
>
> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>
> ...
> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>
> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>
> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>
>
>
>
>

On 02/24/2024 10:20 AM, Ross Finlayson wrote:
> On 02/17/2024 09:24 AM, Ross Finlayson wrote:
>> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>>
>>>>
>>>> So I'm thinking about Factorial/Exponential Identity,
>>>> from 2003, and one of the outputs is like this.
>>>>
>>>> n! ~= root( sum n^n - (sum n) ^n )
>>>>
>>>> Now this amuses me a lot, because, it relates these
>>>> different terms, or rather, in terms of one variable
>>>> a term, by these various operations, relating the
>>>> operations together.
>>>>
>>>> Here n! is n factorial and root is the square root
>>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>>> (n(n+1)/2)^n.
>>>>
>>>> So it's funny because as of a root of a difference,
>>>> it's sort of like a mean
>>>>
>>>> arithmetic mean = n'th-part sum-n-terms
>>>> geometric mean = n'th-root product-n-terms
>>>>
>>>> but just about "roots of difference, of various terms".
>>>>
>>>> Then, ~= means "approximately equal" it's an approximation,
>>>> or "it's true in the limit", here leaving that out leaves
>>>> these kinds of things.
>>>>
>>>> F = n!
>>>> L = sum (n^n)
>>>> R = (sum n)^n = (n(n+1)/2)^n
>>>>
>>>> The most simple sort of algebraic manipulation
>>>> leads for these kinds of things.
>>>>
>>>> F^2 = L - R
>>>>
>>>> L = F^2 + R
>>>>
>>>> R = L - F^2
>>>>
>>>> 2^n (L - F^2) = (n(n+1))^n
>>>>
>>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>>
>>>> n'th-root (2^n (L - F^2)) = X
>>>>
>>>>
>>>> n^2 + n - X = 0
>>>>
>>>> Now I'm used to seeing something like
>>>>
>>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>>
>>>> which gets me to wondering that
>>>>
>>>> X = -1/4
>>>>
>>>> when, "n is a root of this quadratic".
>>>>
>>>> n'th-root (2^n (L - F^2)) = -1/4
>>>>
>>>> (-1/4)^n = 2^n (L - F^2)
>>>>
>>>> (-1/8)^n = L - F^2
>>>>
>>>> (-1/8)^n = sum (n^n) - n!^2
>>>>
>>>> Borel versus Combinatorics, anyone?
>>>>
>>>> Fun with Euler? In the limit, ....
>>>>
>>>> Yet, for keeping things real, is for something like
>>>>
>>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>>
>>>> = n^2 + n - ab, or for where
>>>>
>>>> X = ab = a(a+1) = a^2 + a
>>>>
>>>> when, "n is a root of this quadratic".
>>>>
>>>> n'th-root (2^n (L - F^2)) = ab
>>>>
>>>> (ab)^n = 2^n (L - F^2)
>>>>
>>>> (ab/2)^n = L - F^2
>>>>
>>>> (ab/2)^n = sum (n^n) - n!^2
>>>>
>>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>>
>>>> Borel versus Combinatorics, anyone?
>>>>
>>>> Fun with Euler? In the limit, ....
>>>>
>>>> ((a^2) + a)/2) = A
>>>>
>>>> A^n = L - F^2
>>>>
>>>> A^n/L = 1 - F^2/L
>>>>
>>>> Hmm...
>>>>
>>>> So, looking for a(a+1) = 1,
>>>>
>>>> a^2 + a - 1 = 0
>>>>
>>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>>
>>>> (for b^2 -4ac/2a, ....)
>>>>
>>>> a = (root 5 - 1)/2 ~ 0.618
>>>> a = (-root5 -1)/2 ~= -1.618
>>>>
>>>>
>>>>
>>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>>
>>>> Hmm, you know they say phi the Golden Mean shows up
>>>> everywhere, but, the "negative Golden Mean"?
>>>>
>>>> Then this, "Golden mean minus one"?
>>>>
>>>> So, calling those -phi and phi-1,
>>>>
>>>> -phi:
>>>>
>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>
>>>> n^2 +n + phi^2 - phi = 0
>>>>
>>>> phi-1:
>>>>
>>>> n^2 + n + phi^2 = 0
>>>>
>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>
>>>> Keeping things positive,
>>>>
>>>> phi-1:
>>>>
>>>> phi^2 = X
>>>> phi^2 = n'th-root (2^n (L - F^2))
>>>>
>>>> Great, now it looks like I added the Golden Mean among
>>>> the "Factorial/Exponential Identities" given some various
>>>> ideas in laws of large numbers, a square compact Cantor space,
>>>> some expectations in probability, and some sorts these other things.
>>>>
>>>> Borel vs. combinatorics, anyone?
>>>
>>>
>>>
>>>
>>> Well, whuh, the phi-1 case, ...
>>>
>>> n^2 + n + (phi-1)(phi) = 0
>>>
>>> looks the same as the -phi case, about that,
>>>
>>> -phi:
>>> n^2 + n + (-phi)(-phi+1) = 0
>>> n^2 + n + phi^2 - phi = 0
>>>
>>> phi-1:
>>> n^2 + n + (phi-1)(phi) = 0
>>> n^2 + n + phi^2 - phi = 0
>>>
>>> It looks in both cases that
>>>
>>> phi^2 - phi = X
>>>
>>> X = n'th-root (2^n (L - F^2))
>>>
>>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>
>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>>
>>> Because, phi-1 = 1/phi,
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>
>>>
>>> 1/phi:
>>>
>>> n^2 + n + (1/phi)(1/phi+1) = 0
>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>>> n^2 + n + ((1 +phi)/phi^2) = 0
>>>
>>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>>
>>> X = phi^2, when, "n is a root of this quadratic".
>>>
>>> -1 -1/phi:
>>>
>>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>>> n^2 + n + 1/phi + 1/phi^2 = 0
>>> 1/phi + 1/phi^2 = 1
>>>
>>>
>>> Hmm, phi: came and went, yielded:
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>
>>> So now I'm looking at the Lucas number,
>>> or that "phi = 1 + 1/phi can be expanded
>>> recursively to obtain a continued fraction
>>> for the golden ratio".
>>>
>>> This is where basically is for getting out
>>> the "infinity'th root" of phi, then to put it in.
>>>
>>> "... one can easily decompose any power of phi into
>>> a multiple of phi and a constant. The multiple
>>> and the constant are always adjacent Fibonacci numbers."
>>>
>>> So I'm looking for the "negative Lucas and Fibonacci
>>> numbers", that phi^n has an expression in L_n and F_n,
>>> looking for phi^{-n}.
>>>
>>> "That pi times root phi is _close_ to 4 is only called
>>> 'numerical coincidence'."
>>>
>>> Let's see, if root phi is "the Golden Root", then,
>>> looking for powers of the Golden Root, ....
>>>
>>>
>>> Hmm, phi: came and went, yielded:
>>>
>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>
>>> Yet, any sort expression n'th powers of phi goes in, ....
>>>
>>> X = phi^2 - phi
>>>
>>> phi^2 - phi - X = 0
>>>
>>>
>>>
>>>
>>
>>
>>
>> The Wiki for phi says an interesting thing, about
>> the consistently small terms in its contined fraction,
>> as explain why the approximates to phi^{-1} converge
>> so slowly: that "this makes the golden ratio an extreme
>> case of the Hurwitz inequality for Diophantine approximations."
>>
>>
>> So looking around some guy says Binet's formulas are like so:
>>
>> Lucas number L_n = phi^n + (-phi)^{-n}
>>
>> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>>
>> Then, I'm looking for phi^{-n}, and n goes to infinity,
>> basically for what are these "negative Lucas and Fibonacci
>> numbers", L_-n and F_-n, and what are these "inverse Lucas
>> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
>> looking for phi^{1/n}.
>>
>>
>>
>> "The ratios between consecutive Fibonacci numbers
>> approaches phi...", in the limit as n -> \infty.
>>
>> I suppose the roots in binary are the same as
>> the roots in "phi-nary". I.e., in their representations.
>>
>> "A root-phi rectangle divides into a pair
>> of Kepler triangles (right triangles with
>> edge lengths in geometric progression)."
>> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>>
>> https://en.wikipedia.org/wiki/Kepler_triangle
>>
>> 1^2 + root-phi ^2 = phi^2
>>
>> root phi = root (phi^2 - 1)
>>
>> "The three edge length 1, root phi, and phi are
>> the harmonic mean, geometric mean, and arithmetic mean,
>> respectively, of the two numbers phi +- 1."
>>
>>
>> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>>
>>
>> About 25/4 and 5/2, ....
>>
>> https://mathworld.wolfram.com/GoldenRatio.html
>>
>> phi = 2 cos (pi/5)
>>
>> Not much to be found about "roots of phi".
>> Or, there's tons to be found "roots of phi" historically,
>> not so much analytically.
>>
>> https://arxiv.org/abs/2312.10767 :
>>
>> "Interestingly, in the non-rotating limit, RFD bifurcates
>> into the usual, non-rotating FD branch and into a spurious
>> branch, named "golden" branch, mapping a non-rotating
>> (static) extremal BH to an under-rotating (stationary)
>> extremal BH, in which the ratio between the angular
>> momentum and the non-rotating entropy is the square
>> root of the golden ratio."
>>
>> https://arxiv.org/abs/2210.13395
>>
>> Found some interest Nigerian papers about
>> "inverse powers of phi", phi^{-n}, but looking
>> for "roots of phi", phi^{1/n}.
>>
>>
>> https://arxiv.org/abs/1512.00379
>>
>> "Psi is the square root of the golden ratio."
>>
>> Lakshmi Roychowdhury, Graf and Luschgy,
>> https://link.springer.com/book/10.1007/BFb0103945
>> still looking for "n'th roots of phi".
>>
>> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>>
>> phi = 2 cos (pi/5)
>> phi/2 = cos(pi/5)
>>
>> nth-root phi/2 = nth-root cos(pi/5)
>>
>> 1/phi = phi - 1
>> phi + 1 - 1 = 2 cos (pi/5)
>> 1/phi + 1 = 2 cos (pi/5)
>> 1/phi = 2 cos (pi/5) - 1
>>
>> phi = 1/ ( 2cos (pi/5) - 1
>> phi/2 = cos (pi/5)
>> arccos (phi/2) = pi/5
>> pi = 5 arccos (phi/2)
>>
>> arccos(-phi/2) = 0
>> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
>> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>>
>> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
>> sin(pi/5) = root (1- (phi/2)^2 )
>>
>> cos^2(theta) = cos(pi/5)
>> sin(theta) = +- root( 1-cos(pi/5) )
>> sin(theta) = += root (1-phi/2)
>> theta = +- arcsin(root(1-phi/2))
>> cos(theta) = root(phi)
>> cos(arcsin(root(1-phi/2))) = root(phi)
>>
>> root phi = 1.27202....
>> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>>
>>
>> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root 5)/2.
>>
>> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
>> Of course that's just modular and doesn't say anything about roots.
>>
>> What I'm looking at is:
>>
>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>
>> with the idea to write phi^2 - phi in terms of an infinite series,
>> then get all the terms on one side. Then what I figure is that
>> it would be in terms of the n'th root or to-the-1-over-n,
>> getting those on one side, or just the phi part
>>
>> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>
>> ...
>> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>
>> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>>
>> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>>
>>
>>
>>
>>
>
>
>
> Well I got started figuring out some half-angle formulas,
> with regards to "computing fractional roots of phi"
> with regards to "computing fractional roots of theta"
> with regards to theta being the angle and theta being phi,
> using trigonometry.
>
> Then what I've been looking at is, "infinite series". Euh...,
> infinite series gets involved the "telescoping", and, the
> "scaffolding", with respect to sorts, "reference series",
> in terms of summations and production, "sharing a
> common reference index", that sort of establishing
> telescoping, then, when that collapses down into
> "reference index", for an infinite series, that here
> is prototyped by the inverse powers of two, what
> also happens to be a geometric series, sum of which is 1.
>
> So, I'm looking at such things as deconstructive accounts
> of the geometric series, and a lot that goes on with means,
> with regards to infinite series, reading about Euler and
> this kind of thing, infinite series, with regards to figuring
> out how to help make that the quantities involved,
> result when arithmetic series can or can't have their
> terms grouped and then summed or otherwise
> considered term-wise, infinite series.
>
> I asked around about "do you know any closed forms
> for fractional powers of phi" and it was like "no, and
> I just read a ton of stuff about it", here that what
> I'll be looking at is how to describe the infinite series
> and getting into "surds and adeles" and "the diophantine"
> and "infinite series for e, pi, phi, Catalan's, and so on".
>
>
>
> If you know some particular methodologies about
> infinite series and this kind of thing, I'll be curious
> about establishing what can make for forms, here
> for the "telescoping and scaffolding", what results
> the quantities, result the asymptotics, for various
> law(s) of large numbers, about results like Euler's,
> and Stirling's, then pretty much about continued
> fractions, and infinite series, and addition formula,
> the quantities.
>
>

On 02/24/2024 11:17 AM, Ross Finlayson wrote:
> On 02/24/2024 10:20 AM, Ross Finlayson wrote:
>> On 02/17/2024 09:24 AM, Ross Finlayson wrote:
>>> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>>>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>>>
>>>>>
>>>>> So I'm thinking about Factorial/Exponential Identity,
>>>>> from 2003, and one of the outputs is like this.
>>>>>
>>>>> n! ~= root( sum n^n - (sum n) ^n )
>>>>>
>>>>> Now this amuses me a lot, because, it relates these
>>>>> different terms, or rather, in terms of one variable
>>>>> a term, by these various operations, relating the
>>>>> operations together.
>>>>>
>>>>> Here n! is n factorial and root is the square root
>>>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>>>> (n(n+1)/2)^n.
>>>>>
>>>>> So it's funny because as of a root of a difference,
>>>>> it's sort of like a mean
>>>>>
>>>>> arithmetic mean = n'th-part sum-n-terms
>>>>> geometric mean = n'th-root product-n-terms
>>>>>
>>>>> but just about "roots of difference, of various terms".
>>>>>
>>>>> Then, ~= means "approximately equal" it's an approximation,
>>>>> or "it's true in the limit", here leaving that out leaves
>>>>> these kinds of things.
>>>>>
>>>>> F = n!
>>>>> L = sum (n^n)
>>>>> R = (sum n)^n = (n(n+1)/2)^n
>>>>>
>>>>> The most simple sort of algebraic manipulation
>>>>> leads for these kinds of things.
>>>>>
>>>>> F^2 = L - R
>>>>>
>>>>> L = F^2 + R
>>>>>
>>>>> R = L - F^2
>>>>>
>>>>> 2^n (L - F^2) = (n(n+1))^n
>>>>>
>>>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>>>
>>>>> n'th-root (2^n (L - F^2)) = X
>>>>>
>>>>>
>>>>> n^2 + n - X = 0
>>>>>
>>>>> Now I'm used to seeing something like
>>>>>
>>>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>>>
>>>>> which gets me to wondering that
>>>>>
>>>>> X = -1/4
>>>>>
>>>>> when, "n is a root of this quadratic".
>>>>>
>>>>> n'th-root (2^n (L - F^2)) = -1/4
>>>>>
>>>>> (-1/4)^n = 2^n (L - F^2)
>>>>>
>>>>> (-1/8)^n = L - F^2
>>>>>
>>>>> (-1/8)^n = sum (n^n) - n!^2
>>>>>
>>>>> Borel versus Combinatorics, anyone?
>>>>>
>>>>> Fun with Euler? In the limit, ....
>>>>>
>>>>> Yet, for keeping things real, is for something like
>>>>>
>>>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>>>
>>>>> = n^2 + n - ab, or for where
>>>>>
>>>>> X = ab = a(a+1) = a^2 + a
>>>>>
>>>>> when, "n is a root of this quadratic".
>>>>>
>>>>> n'th-root (2^n (L - F^2)) = ab
>>>>>
>>>>> (ab)^n = 2^n (L - F^2)
>>>>>
>>>>> (ab/2)^n = L - F^2
>>>>>
>>>>> (ab/2)^n = sum (n^n) - n!^2
>>>>>
>>>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>>>
>>>>> Borel versus Combinatorics, anyone?
>>>>>
>>>>> Fun with Euler? In the limit, ....
>>>>>
>>>>> ((a^2) + a)/2) = A
>>>>>
>>>>> A^n = L - F^2
>>>>>
>>>>> A^n/L = 1 - F^2/L
>>>>>
>>>>> Hmm...
>>>>>
>>>>> So, looking for a(a+1) = 1,
>>>>>
>>>>> a^2 + a - 1 = 0
>>>>>
>>>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>>>
>>>>> (for b^2 -4ac/2a, ....)
>>>>>
>>>>> a = (root 5 - 1)/2 ~ 0.618
>>>>> a = (-root5 -1)/2 ~= -1.618
>>>>>
>>>>>
>>>>>
>>>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>>>
>>>>> Hmm, you know they say phi the Golden Mean shows up
>>>>> everywhere, but, the "negative Golden Mean"?
>>>>>
>>>>> Then this, "Golden mean minus one"?
>>>>>
>>>>> So, calling those -phi and phi-1,
>>>>>
>>>>> -phi:
>>>>>
>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>
>>>>> n^2 +n + phi^2 - phi = 0
>>>>>
>>>>> phi-1:
>>>>>
>>>>> n^2 + n + phi^2 = 0
>>>>>
>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>
>>>>> Keeping things positive,
>>>>>
>>>>> phi-1:
>>>>>
>>>>> phi^2 = X
>>>>> phi^2 = n'th-root (2^n (L - F^2))
>>>>>
>>>>> Great, now it looks like I added the Golden Mean among
>>>>> the "Factorial/Exponential Identities" given some various
>>>>> ideas in laws of large numbers, a square compact Cantor space,
>>>>> some expectations in probability, and some sorts these other things.
>>>>>
>>>>> Borel vs. combinatorics, anyone?
>>>>
>>>>
>>>>
>>>>
>>>> Well, whuh, the phi-1 case, ...
>>>>
>>>> n^2 + n + (phi-1)(phi) = 0
>>>>
>>>> looks the same as the -phi case, about that,
>>>>
>>>> -phi:
>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>> n^2 + n + phi^2 - phi = 0
>>>>
>>>> phi-1:
>>>> n^2 + n + (phi-1)(phi) = 0
>>>> n^2 + n + phi^2 - phi = 0
>>>>
>>>> It looks in both cases that
>>>>
>>>> phi^2 - phi = X
>>>>
>>>> X = n'th-root (2^n (L - F^2))
>>>>
>>>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>
>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>>>
>>>> Because, phi-1 = 1/phi,
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>
>>>>
>>>> 1/phi:
>>>>
>>>> n^2 + n + (1/phi)(1/phi+1) = 0
>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>>>> n^2 + n + ((1 +phi)/phi^2) = 0
>>>>
>>>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>>>
>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>
>>>> -1 -1/phi:
>>>>
>>>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>>>> n^2 + n + 1/phi + 1/phi^2 = 0
>>>> 1/phi + 1/phi^2 = 1
>>>>
>>>>
>>>> Hmm, phi: came and went, yielded:
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>
>>>> So now I'm looking at the Lucas number,
>>>> or that "phi = 1 + 1/phi can be expanded
>>>> recursively to obtain a continued fraction
>>>> for the golden ratio".
>>>>
>>>> This is where basically is for getting out
>>>> the "infinity'th root" of phi, then to put it in.
>>>>
>>>> "... one can easily decompose any power of phi into
>>>> a multiple of phi and a constant. The multiple
>>>> and the constant are always adjacent Fibonacci numbers."
>>>>
>>>> So I'm looking for the "negative Lucas and Fibonacci
>>>> numbers", that phi^n has an expression in L_n and F_n,
>>>> looking for phi^{-n}.
>>>>
>>>> "That pi times root phi is _close_ to 4 is only called
>>>> 'numerical coincidence'."
>>>>
>>>> Let's see, if root phi is "the Golden Root", then,
>>>> looking for powers of the Golden Root, ....
>>>>
>>>>
>>>> Hmm, phi: came and went, yielded:
>>>>
>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>
>>>> Yet, any sort expression n'th powers of phi goes in, ....
>>>>
>>>> X = phi^2 - phi
>>>>
>>>> phi^2 - phi - X = 0
>>>>
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>> The Wiki for phi says an interesting thing, about
>>> the consistently small terms in its contined fraction,
>>> as explain why the approximates to phi^{-1} converge
>>> so slowly: that "this makes the golden ratio an extreme
>>> case of the Hurwitz inequality for Diophantine approximations."
>>>
>>>
>>> So looking around some guy says Binet's formulas are like so:
>>>
>>> Lucas number L_n = phi^n + (-phi)^{-n}
>>>
>>> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>>>
>>> Then, I'm looking for phi^{-n}, and n goes to infinity,
>>> basically for what are these "negative Lucas and Fibonacci
>>> numbers", L_-n and F_-n, and what are these "inverse Lucas
>>> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
>>> looking for phi^{1/n}.
>>>
>>>
>>>
>>> "The ratios between consecutive Fibonacci numbers
>>> approaches phi...", in the limit as n -> \infty.
>>>
>>> I suppose the roots in binary are the same as
>>> the roots in "phi-nary". I.e., in their representations.
>>>
>>> "A root-phi rectangle divides into a pair
>>> of Kepler triangles (right triangles with
>>> edge lengths in geometric progression)."
>>> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>>>
>>> https://en.wikipedia.org/wiki/Kepler_triangle
>>>
>>> 1^2 + root-phi ^2 = phi^2
>>>
>>> root phi = root (phi^2 - 1)
>>>
>>> "The three edge length 1, root phi, and phi are
>>> the harmonic mean, geometric mean, and arithmetic mean,
>>> respectively, of the two numbers phi +- 1."
>>>
>>>
>>> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>>>
>>>
>>> About 25/4 and 5/2, ....
>>>
>>> https://mathworld.wolfram.com/GoldenRatio.html
>>>
>>> phi = 2 cos (pi/5)
>>>
>>> Not much to be found about "roots of phi".
>>> Or, there's tons to be found "roots of phi" historically,
>>> not so much analytically.
>>>
>>> https://arxiv.org/abs/2312.10767 :
>>>
>>> "Interestingly, in the non-rotating limit, RFD bifurcates
>>> into the usual, non-rotating FD branch and into a spurious
>>> branch, named "golden" branch, mapping a non-rotating
>>> (static) extremal BH to an under-rotating (stationary)
>>> extremal BH, in which the ratio between the angular
>>> momentum and the non-rotating entropy is the square
>>> root of the golden ratio."
>>>
>>> https://arxiv.org/abs/2210.13395
>>>
>>> Found some interest Nigerian papers about
>>> "inverse powers of phi", phi^{-n}, but looking
>>> for "roots of phi", phi^{1/n}.
>>>
>>>
>>> https://arxiv.org/abs/1512.00379
>>>
>>> "Psi is the square root of the golden ratio."
>>>
>>> Lakshmi Roychowdhury, Graf and Luschgy,
>>> https://link.springer.com/book/10.1007/BFb0103945
>>> still looking for "n'th roots of phi".
>>>
>>> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>>>
>>> phi = 2 cos (pi/5)
>>> phi/2 = cos(pi/5)
>>>
>>> nth-root phi/2 = nth-root cos(pi/5)
>>>
>>> 1/phi = phi - 1
>>> phi + 1 - 1 = 2 cos (pi/5)
>>> 1/phi + 1 = 2 cos (pi/5)
>>> 1/phi = 2 cos (pi/5) - 1
>>>
>>> phi = 1/ ( 2cos (pi/5) - 1
>>> phi/2 = cos (pi/5)
>>> arccos (phi/2) = pi/5
>>> pi = 5 arccos (phi/2)
>>>
>>> arccos(-phi/2) = 0
>>> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
>>> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>>>
>>> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
>>> sin(pi/5) = root (1- (phi/2)^2 )
>>>
>>> cos^2(theta) = cos(pi/5)
>>> sin(theta) = +- root( 1-cos(pi/5) )
>>> sin(theta) = += root (1-phi/2)
>>> theta = +- arcsin(root(1-phi/2))
>>> cos(theta) = root(phi)
>>> cos(arcsin(root(1-phi/2))) = root(phi)
>>>
>>> root phi = 1.27202....
>>> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>>>
>>>
>>> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root
>>> 5)/2.
>>>
>>> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
>>> Of course that's just modular and doesn't say anything about roots.
>>>
>>> What I'm looking at is:
>>>
>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>
>>> with the idea to write phi^2 - phi in terms of an infinite series,
>>> then get all the terms on one side. Then what I figure is that
>>> it would be in terms of the n'th root or to-the-1-over-n,
>>> getting those on one side, or just the phi part
>>>
>>> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>
>>> ...
>>> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>
>>> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>>>
>>> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>>>
>>>
>>>
>>>
>>>
>>
>>
>>
>> Well I got started figuring out some half-angle formulas,
>> with regards to "computing fractional roots of phi"
>> with regards to "computing fractional roots of theta"
>> with regards to theta being the angle and theta being phi,
>> using trigonometry.
>>
>> Then what I've been looking at is, "infinite series". Euh...,
>> infinite series gets involved the "telescoping", and, the
>> "scaffolding", with respect to sorts, "reference series",
>> in terms of summations and production, "sharing a
>> common reference index", that sort of establishing
>> telescoping, then, when that collapses down into
>> "reference index", for an infinite series, that here
>> is prototyped by the inverse powers of two, what
>> also happens to be a geometric series, sum of which is 1.
>>
>> So, I'm looking at such things as deconstructive accounts
>> of the geometric series, and a lot that goes on with means,
>> with regards to infinite series, reading about Euler and
>> this kind of thing, infinite series, with regards to figuring
>> out how to help make that the quantities involved,
>> result when arithmetic series can or can't have their
>> terms grouped and then summed or otherwise
>> considered term-wise, infinite series.
>>
>> I asked around about "do you know any closed forms
>> for fractional powers of phi" and it was like "no, and
>> I just read a ton of stuff about it", here that what
>> I'll be looking at is how to describe the infinite series
>> and getting into "surds and adeles" and "the diophantine"
>> and "infinite series for e, pi, phi, Catalan's, and so on".
>>
>>
>>
>> If you know some particular methodologies about
>> infinite series and this kind of thing, I'll be curious
>> about establishing what can make for forms, here
>> for the "telescoping and scaffolding", what results
>> the quantities, result the asymptotics, for various
>> law(s) of large numbers, about results like Euler's,
>> and Stirling's, then pretty much about continued
>> fractions, and infinite series, and addition formula,
>> the quantities.
>>
>>
>
>
>
> Continued fractions and phi have something
> together where it's [1, 1, 1, 1, ...], about things
> like [0, 1, 1, 1, 1, ...], and so on, vis-a-vis infinite
> series and things like
>
> +++++...
> +-+-+-+...
>
> and the idea that "infinite series must have a
> point at infinity" then with regards to "also
> infinite series oscillate and indicate all the
> way out to infinity".
>
> There's infinite series then there's infinite products,
> about all the great utility of "addition formulas"
> and getting into "e^x + e^-x" and "sums and differences
> of roots and powers and roots and powers of sums
> and differences", "addition formulas" and "reciprocity".
>
> Then this idea of "reference index" and "indexicality",
> as well gets into the "infinite metric", about how
> the sum of the inverse powers of two equals one,
> and very specific references series and products
> for the most reduced, as it were, constants, of mathematics.
>
> https://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan
> https://en.wikipedia.org/wiki/Lorenzo_Mascheroni
>
> https://en.wikipedia.org/wiki/James_Stirling_(mathematician)
> https://en.wikipedia.org/wiki/Abraham_de_Moivre
>
> Submitted, another _belle lettre_ on the stack.
>
>

On 03/04/2024 10:53 AM, Ross Finlayson wrote:
> On 02/24/2024 11:17 AM, Ross Finlayson wrote:
>> On 02/24/2024 10:20 AM, Ross Finlayson wrote:
>>> On 02/17/2024 09:24 AM, Ross Finlayson wrote:
>>>> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>>>>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>>>>
>>>>>>
>>>>>> So I'm thinking about Factorial/Exponential Identity,
>>>>>> from 2003, and one of the outputs is like this.
>>>>>>
>>>>>> n! ~= root( sum n^n - (sum n) ^n )
>>>>>>
>>>>>> Now this amuses me a lot, because, it relates these
>>>>>> different terms, or rather, in terms of one variable
>>>>>> a term, by these various operations, relating the
>>>>>> operations together.
>>>>>>
>>>>>> Here n! is n factorial and root is the square root
>>>>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>>>>> (n(n+1)/2)^n.
>>>>>>
>>>>>> So it's funny because as of a root of a difference,
>>>>>> it's sort of like a mean
>>>>>>
>>>>>> arithmetic mean = n'th-part sum-n-terms
>>>>>> geometric mean = n'th-root product-n-terms
>>>>>>
>>>>>> but just about "roots of difference, of various terms".
>>>>>>
>>>>>> Then, ~= means "approximately equal" it's an approximation,
>>>>>> or "it's true in the limit", here leaving that out leaves
>>>>>> these kinds of things.
>>>>>>
>>>>>> F = n!
>>>>>> L = sum (n^n)
>>>>>> R = (sum n)^n = (n(n+1)/2)^n
>>>>>>
>>>>>> The most simple sort of algebraic manipulation
>>>>>> leads for these kinds of things.
>>>>>>
>>>>>> F^2 = L - R
>>>>>>
>>>>>> L = F^2 + R
>>>>>>
>>>>>> R = L - F^2
>>>>>>
>>>>>> 2^n (L - F^2) = (n(n+1))^n
>>>>>>
>>>>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>>>>
>>>>>> n'th-root (2^n (L - F^2)) = X
>>>>>>
>>>>>>
>>>>>> n^2 + n - X = 0
>>>>>>
>>>>>> Now I'm used to seeing something like
>>>>>>
>>>>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>>>>
>>>>>> which gets me to wondering that
>>>>>>
>>>>>> X = -1/4
>>>>>>
>>>>>> when, "n is a root of this quadratic".
>>>>>>
>>>>>> n'th-root (2^n (L - F^2)) = -1/4
>>>>>>
>>>>>> (-1/4)^n = 2^n (L - F^2)
>>>>>>
>>>>>> (-1/8)^n = L - F^2
>>>>>>
>>>>>> (-1/8)^n = sum (n^n) - n!^2
>>>>>>
>>>>>> Borel versus Combinatorics, anyone?
>>>>>>
>>>>>> Fun with Euler? In the limit, ....
>>>>>>
>>>>>> Yet, for keeping things real, is for something like
>>>>>>
>>>>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>>>>
>>>>>> = n^2 + n - ab, or for where
>>>>>>
>>>>>> X = ab = a(a+1) = a^2 + a
>>>>>>
>>>>>> when, "n is a root of this quadratic".
>>>>>>
>>>>>> n'th-root (2^n (L - F^2)) = ab
>>>>>>
>>>>>> (ab)^n = 2^n (L - F^2)
>>>>>>
>>>>>> (ab/2)^n = L - F^2
>>>>>>
>>>>>> (ab/2)^n = sum (n^n) - n!^2
>>>>>>
>>>>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>>>>
>>>>>> Borel versus Combinatorics, anyone?
>>>>>>
>>>>>> Fun with Euler? In the limit, ....
>>>>>>
>>>>>> ((a^2) + a)/2) = A
>>>>>>
>>>>>> A^n = L - F^2
>>>>>>
>>>>>> A^n/L = 1 - F^2/L
>>>>>>
>>>>>> Hmm...
>>>>>>
>>>>>> So, looking for a(a+1) = 1,
>>>>>>
>>>>>> a^2 + a - 1 = 0
>>>>>>
>>>>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>>>>
>>>>>> (for b^2 -4ac/2a, ....)
>>>>>>
>>>>>> a = (root 5 - 1)/2 ~ 0.618
>>>>>> a = (-root5 -1)/2 ~= -1.618
>>>>>>
>>>>>>
>>>>>>
>>>>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>>>>
>>>>>> Hmm, you know they say phi the Golden Mean shows up
>>>>>> everywhere, but, the "negative Golden Mean"?
>>>>>>
>>>>>> Then this, "Golden mean minus one"?
>>>>>>
>>>>>> So, calling those -phi and phi-1,
>>>>>>
>>>>>> -phi:
>>>>>>
>>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>>
>>>>>> n^2 +n + phi^2 - phi = 0
>>>>>>
>>>>>> phi-1:
>>>>>>
>>>>>> n^2 + n + phi^2 = 0
>>>>>>
>>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>>
>>>>>> Keeping things positive,
>>>>>>
>>>>>> phi-1:
>>>>>>
>>>>>> phi^2 = X
>>>>>> phi^2 = n'th-root (2^n (L - F^2))
>>>>>>
>>>>>> Great, now it looks like I added the Golden Mean among
>>>>>> the "Factorial/Exponential Identities" given some various
>>>>>> ideas in laws of large numbers, a square compact Cantor space,
>>>>>> some expectations in probability, and some sorts these other things.
>>>>>>
>>>>>> Borel vs. combinatorics, anyone?
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> Well, whuh, the phi-1 case, ...
>>>>>
>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>>
>>>>> looks the same as the -phi case, about that,
>>>>>
>>>>> -phi:
>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>> n^2 + n + phi^2 - phi = 0
>>>>>
>>>>> phi-1:
>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>> n^2 + n + phi^2 - phi = 0
>>>>>
>>>>> It looks in both cases that
>>>>>
>>>>> phi^2 - phi = X
>>>>>
>>>>> X = n'th-root (2^n (L - F^2))
>>>>>
>>>>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>
>>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>>>>
>>>>> Because, phi-1 = 1/phi,
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>
>>>>>
>>>>> 1/phi:
>>>>>
>>>>> n^2 + n + (1/phi)(1/phi+1) = 0
>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>>>>> n^2 + n + ((1 +phi)/phi^2) = 0
>>>>>
>>>>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>>>>
>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>
>>>>> -1 -1/phi:
>>>>>
>>>>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>>>>> n^2 + n + 1/phi + 1/phi^2 = 0
>>>>> 1/phi + 1/phi^2 = 1
>>>>>
>>>>>
>>>>> Hmm, phi: came and went, yielded:
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>
>>>>> So now I'm looking at the Lucas number,
>>>>> or that "phi = 1 + 1/phi can be expanded
>>>>> recursively to obtain a continued fraction
>>>>> for the golden ratio".
>>>>>
>>>>> This is where basically is for getting out
>>>>> the "infinity'th root" of phi, then to put it in.
>>>>>
>>>>> "... one can easily decompose any power of phi into
>>>>> a multiple of phi and a constant. The multiple
>>>>> and the constant are always adjacent Fibonacci numbers."
>>>>>
>>>>> So I'm looking for the "negative Lucas and Fibonacci
>>>>> numbers", that phi^n has an expression in L_n and F_n,
>>>>> looking for phi^{-n}.
>>>>>
>>>>> "That pi times root phi is _close_ to 4 is only called
>>>>> 'numerical coincidence'."
>>>>>
>>>>> Let's see, if root phi is "the Golden Root", then,
>>>>> looking for powers of the Golden Root, ....
>>>>>
>>>>>
>>>>> Hmm, phi: came and went, yielded:
>>>>>
>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>
>>>>> Yet, any sort expression n'th powers of phi goes in, ....
>>>>>
>>>>> X = phi^2 - phi
>>>>>
>>>>> phi^2 - phi - X = 0
>>>>>
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>> The Wiki for phi says an interesting thing, about
>>>> the consistently small terms in its contined fraction,
>>>> as explain why the approximates to phi^{-1} converge
>>>> so slowly: that "this makes the golden ratio an extreme
>>>> case of the Hurwitz inequality for Diophantine approximations."
>>>>
>>>>
>>>> So looking around some guy says Binet's formulas are like so:
>>>>
>>>> Lucas number L_n = phi^n + (-phi)^{-n}
>>>>
>>>> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>>>>
>>>> Then, I'm looking for phi^{-n}, and n goes to infinity,
>>>> basically for what are these "negative Lucas and Fibonacci
>>>> numbers", L_-n and F_-n, and what are these "inverse Lucas
>>>> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
>>>> looking for phi^{1/n}.
>>>>
>>>>
>>>>
>>>> "The ratios between consecutive Fibonacci numbers
>>>> approaches phi...", in the limit as n -> \infty.
>>>>
>>>> I suppose the roots in binary are the same as
>>>> the roots in "phi-nary". I.e., in their representations.
>>>>
>>>> "A root-phi rectangle divides into a pair
>>>> of Kepler triangles (right triangles with
>>>> edge lengths in geometric progression)."
>>>> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>>>>
>>>> https://en.wikipedia.org/wiki/Kepler_triangle
>>>>
>>>> 1^2 + root-phi ^2 = phi^2
>>>>
>>>> root phi = root (phi^2 - 1)
>>>>
>>>> "The three edge length 1, root phi, and phi are
>>>> the harmonic mean, geometric mean, and arithmetic mean,
>>>> respectively, of the two numbers phi +- 1."
>>>>
>>>>
>>>> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>>>>
>>>>
>>>> About 25/4 and 5/2, ....
>>>>
>>>> https://mathworld.wolfram.com/GoldenRatio.html
>>>>
>>>> phi = 2 cos (pi/5)
>>>>
>>>> Not much to be found about "roots of phi".
>>>> Or, there's tons to be found "roots of phi" historically,
>>>> not so much analytically.
>>>>
>>>> https://arxiv.org/abs/2312.10767 :
>>>>
>>>> "Interestingly, in the non-rotating limit, RFD bifurcates
>>>> into the usual, non-rotating FD branch and into a spurious
>>>> branch, named "golden" branch, mapping a non-rotating
>>>> (static) extremal BH to an under-rotating (stationary)
>>>> extremal BH, in which the ratio between the angular
>>>> momentum and the non-rotating entropy is the square
>>>> root of the golden ratio."
>>>>
>>>> https://arxiv.org/abs/2210.13395
>>>>
>>>> Found some interest Nigerian papers about
>>>> "inverse powers of phi", phi^{-n}, but looking
>>>> for "roots of phi", phi^{1/n}.
>>>>
>>>>
>>>> https://arxiv.org/abs/1512.00379
>>>>
>>>> "Psi is the square root of the golden ratio."
>>>>
>>>> Lakshmi Roychowdhury, Graf and Luschgy,
>>>> https://link.springer.com/book/10.1007/BFb0103945
>>>> still looking for "n'th roots of phi".
>>>>
>>>> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>>>>
>>>> phi = 2 cos (pi/5)
>>>> phi/2 = cos(pi/5)
>>>>
>>>> nth-root phi/2 = nth-root cos(pi/5)
>>>>
>>>> 1/phi = phi - 1
>>>> phi + 1 - 1 = 2 cos (pi/5)
>>>> 1/phi + 1 = 2 cos (pi/5)
>>>> 1/phi = 2 cos (pi/5) - 1
>>>>
>>>> phi = 1/ ( 2cos (pi/5) - 1
>>>> phi/2 = cos (pi/5)
>>>> arccos (phi/2) = pi/5
>>>> pi = 5 arccos (phi/2)
>>>>
>>>> arccos(-phi/2) = 0
>>>> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
>>>> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>>>>
>>>> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
>>>> sin(pi/5) = root (1- (phi/2)^2 )
>>>>
>>>> cos^2(theta) = cos(pi/5)
>>>> sin(theta) = +- root( 1-cos(pi/5) )
>>>> sin(theta) = += root (1-phi/2)
>>>> theta = +- arcsin(root(1-phi/2))
>>>> cos(theta) = root(phi)
>>>> cos(arcsin(root(1-phi/2))) = root(phi)
>>>>
>>>> root phi = 1.27202....
>>>> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>>>>
>>>>
>>>> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root
>>>> 5)/2.
>>>>
>>>> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
>>>> Of course that's just modular and doesn't say anything about roots.
>>>>
>>>> What I'm looking at is:
>>>>
>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>
>>>> with the idea to write phi^2 - phi in terms of an infinite series,
>>>> then get all the terms on one side. Then what I figure is that
>>>> it would be in terms of the n'th root or to-the-1-over-n,
>>>> getting those on one side, or just the phi part
>>>>
>>>> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>
>>>> ...
>>>> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>
>>>> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>>>>
>>>> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>>>>
>>>>
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>> Well I got started figuring out some half-angle formulas,
>>> with regards to "computing fractional roots of phi"
>>> with regards to "computing fractional roots of theta"
>>> with regards to theta being the angle and theta being phi,
>>> using trigonometry.
>>>
>>> Then what I've been looking at is, "infinite series". Euh...,
>>> infinite series gets involved the "telescoping", and, the
>>> "scaffolding", with respect to sorts, "reference series",
>>> in terms of summations and production, "sharing a
>>> common reference index", that sort of establishing
>>> telescoping, then, when that collapses down into
>>> "reference index", for an infinite series, that here
>>> is prototyped by the inverse powers of two, what
>>> also happens to be a geometric series, sum of which is 1.
>>>
>>> So, I'm looking at such things as deconstructive accounts
>>> of the geometric series, and a lot that goes on with means,
>>> with regards to infinite series, reading about Euler and
>>> this kind of thing, infinite series, with regards to figuring
>>> out how to help make that the quantities involved,
>>> result when arithmetic series can or can't have their
>>> terms grouped and then summed or otherwise
>>> considered term-wise, infinite series.
>>>
>>> I asked around about "do you know any closed forms
>>> for fractional powers of phi" and it was like "no, and
>>> I just read a ton of stuff about it", here that what
>>> I'll be looking at is how to describe the infinite series
>>> and getting into "surds and adeles" and "the diophantine"
>>> and "infinite series for e, pi, phi, Catalan's, and so on".
>>>
>>>
>>>
>>> If you know some particular methodologies about
>>> infinite series and this kind of thing, I'll be curious
>>> about establishing what can make for forms, here
>>> for the "telescoping and scaffolding", what results
>>> the quantities, result the asymptotics, for various
>>> law(s) of large numbers, about results like Euler's,
>>> and Stirling's, then pretty much about continued
>>> fractions, and infinite series, and addition formula,
>>> the quantities.
>>>
>>>
>>
>>
>>
>> Continued fractions and phi have something
>> together where it's [1, 1, 1, 1, ...], about things
>> like [0, 1, 1, 1, 1, ...], and so on, vis-a-vis infinite
>> series and things like
>>
>> +++++...
>> +-+-+-+...
>>
>> and the idea that "infinite series must have a
>> point at infinity" then with regards to "also
>> infinite series oscillate and indicate all the
>> way out to infinity".
>>
>> There's infinite series then there's infinite products,
>> about all the great utility of "addition formulas"
>> and getting into "e^x + e^-x" and "sums and differences
>> of roots and powers and roots and powers of sums
>> and differences", "addition formulas" and "reciprocity".
>>
>> Then this idea of "reference index" and "indexicality",
>> as well gets into the "infinite metric", about how
>> the sum of the inverse powers of two equals one,
>> and very specific references series and products
>> for the most reduced, as it were, constants, of mathematics.
>>
>> https://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan
>> https://en.wikipedia.org/wiki/Lorenzo_Mascheroni
>>
>> https://en.wikipedia.org/wiki/James_Stirling_(mathematician)
>> https://en.wikipedia.org/wiki/Abraham_de_Moivre
>>
>> Submitted, another _belle lettre_ on the stack.
>>
>>
>
> Today's about n^2 and 2^n.
>
> The other day I'm reading some writings of Frege,
> that were published about fifty years after writing,
> and he's just mentioning some things while talking
> about "ideas" and "thoughts", or as might be framed
> "concepts" and "contingencies", and he wrote:
>
> 2^4 = 4^2
>
> I looked at this for a second and said to myself,
> well I'll be danged, that looks like the last one
> of those in the integers.
>
> For a while I've been looking for a natural mathematical
> constant greater than 4, so I'm wondering about this.
>
> 0+0 = 0*0 ? 0^0
> 1+1 ! 1*1 = 1^1
> 2+2 = 2*2 = 2^2
> 3+3 ! 3*3 ! 3^3
> ....
>
> So I'm looking at 2^4 and 4^2,
> and wondering about x^y = y^x,
> vis-a-vis xy = yx and x+y = y+x.
>
> I sort of recall learning fractional powers in school,
> and the instructor made a special case for 0^0.
> I really appreciated this because he made both cases,
> then explained, that it was a matter of convention
> and otherwise book-keeping, and as we were quite
> already in the midst of learning the integral calculus,
> we had the concepts to consider that it's 0 or 1,
> which is the convention that it's 1.
>
> So now I'm much wondering about x^y = y^x,
> and about the identities in the forms, or,
> "where the forms intercept the identity function",
> here as what's called "the identity dimension".
>
> In fractional powers, and 1/oo = 0, and x^0 = 1,
> gets into "the infinite inverse powers equal 1",
> another way to look at "roots of unity" than
> the usual great circle often introduced in the
> complex analysis, "roots of unity".
>
> So, how does this relate to finding forms for
> fractional powers of phi? It sort of does. The
> idea is that for any x, there exists y, x =/= y,
> such that x^y = y^x, or not.
>
> 2^4 = 4^2
>
> What I'm thinking is that underneath 2, and
> above two, result two sorts regimes with
> respect to the exponential, and about how
> to relate that, with regards to the identity
> dimension x = y, anywhere that results
> splitting the quandrant into octants, and
> having the "surface" their in either octant
> above and below x = y, then expanding those
> to each being quadrants, those being interesting
> new identity dimension surfaces.
>
>
> Then, the idea is that of course exponential
> grows much, much faster than polynomial,
> about something it results that there are
> many, many more roots "inside the box",
> making for forms that help relate things
> like finding forms for fractional powers of phi.
>
>

On 03/04/2024 08:25 PM, Ross Finlayson wrote:
> On 03/04/2024 10:53 AM, Ross Finlayson wrote:
>> On 02/24/2024 11:17 AM, Ross Finlayson wrote:
>>> On 02/24/2024 10:20 AM, Ross Finlayson wrote:
>>>> On 02/17/2024 09:24 AM, Ross Finlayson wrote:
>>>>> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>>>>>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>>>>>
>>>>>>>
>>>>>>> So I'm thinking about Factorial/Exponential Identity,
>>>>>>> from 2003, and one of the outputs is like this.
>>>>>>>
>>>>>>> n! ~= root( sum n^n - (sum n) ^n )
>>>>>>>
>>>>>>> Now this amuses me a lot, because, it relates these
>>>>>>> different terms, or rather, in terms of one variable
>>>>>>> a term, by these various operations, relating the
>>>>>>> operations together.
>>>>>>>
>>>>>>> Here n! is n factorial and root is the square root
>>>>>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>>>>>> (n(n+1)/2)^n.
>>>>>>>
>>>>>>> So it's funny because as of a root of a difference,
>>>>>>> it's sort of like a mean
>>>>>>>
>>>>>>> arithmetic mean = n'th-part sum-n-terms
>>>>>>> geometric mean = n'th-root product-n-terms
>>>>>>>
>>>>>>> but just about "roots of difference, of various terms".
>>>>>>>
>>>>>>> Then, ~= means "approximately equal" it's an approximation,
>>>>>>> or "it's true in the limit", here leaving that out leaves
>>>>>>> these kinds of things.
>>>>>>>
>>>>>>> F = n!
>>>>>>> L = sum (n^n)
>>>>>>> R = (sum n)^n = (n(n+1)/2)^n
>>>>>>>
>>>>>>> The most simple sort of algebraic manipulation
>>>>>>> leads for these kinds of things.
>>>>>>>
>>>>>>> F^2 = L - R
>>>>>>>
>>>>>>> L = F^2 + R
>>>>>>>
>>>>>>> R = L - F^2
>>>>>>>
>>>>>>> 2^n (L - F^2) = (n(n+1))^n
>>>>>>>
>>>>>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>>>>>
>>>>>>> n'th-root (2^n (L - F^2)) = X
>>>>>>>
>>>>>>>
>>>>>>> n^2 + n - X = 0
>>>>>>>
>>>>>>> Now I'm used to seeing something like
>>>>>>>
>>>>>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>>>>>
>>>>>>> which gets me to wondering that
>>>>>>>
>>>>>>> X = -1/4
>>>>>>>
>>>>>>> when, "n is a root of this quadratic".
>>>>>>>
>>>>>>> n'th-root (2^n (L - F^2)) = -1/4
>>>>>>>
>>>>>>> (-1/4)^n = 2^n (L - F^2)
>>>>>>>
>>>>>>> (-1/8)^n = L - F^2
>>>>>>>
>>>>>>> (-1/8)^n = sum (n^n) - n!^2
>>>>>>>
>>>>>>> Borel versus Combinatorics, anyone?
>>>>>>>
>>>>>>> Fun with Euler? In the limit, ....
>>>>>>>
>>>>>>> Yet, for keeping things real, is for something like
>>>>>>>
>>>>>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>>>>>
>>>>>>> = n^2 + n - ab, or for where
>>>>>>>
>>>>>>> X = ab = a(a+1) = a^2 + a
>>>>>>>
>>>>>>> when, "n is a root of this quadratic".
>>>>>>>
>>>>>>> n'th-root (2^n (L - F^2)) = ab
>>>>>>>
>>>>>>> (ab)^n = 2^n (L - F^2)
>>>>>>>
>>>>>>> (ab/2)^n = L - F^2
>>>>>>>
>>>>>>> (ab/2)^n = sum (n^n) - n!^2
>>>>>>>
>>>>>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>>>>>
>>>>>>> Borel versus Combinatorics, anyone?
>>>>>>>
>>>>>>> Fun with Euler? In the limit, ....
>>>>>>>
>>>>>>> ((a^2) + a)/2) = A
>>>>>>>
>>>>>>> A^n = L - F^2
>>>>>>>
>>>>>>> A^n/L = 1 - F^2/L
>>>>>>>
>>>>>>> Hmm...
>>>>>>>
>>>>>>> So, looking for a(a+1) = 1,
>>>>>>>
>>>>>>> a^2 + a - 1 = 0
>>>>>>>
>>>>>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>>>>>
>>>>>>> (for b^2 -4ac/2a, ....)
>>>>>>>
>>>>>>> a = (root 5 - 1)/2 ~ 0.618
>>>>>>> a = (-root5 -1)/2 ~= -1.618
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>>>>>
>>>>>>> Hmm, you know they say phi the Golden Mean shows up
>>>>>>> everywhere, but, the "negative Golden Mean"?
>>>>>>>
>>>>>>> Then this, "Golden mean minus one"?
>>>>>>>
>>>>>>> So, calling those -phi and phi-1,
>>>>>>>
>>>>>>> -phi:
>>>>>>>
>>>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>>>
>>>>>>> n^2 +n + phi^2 - phi = 0
>>>>>>>
>>>>>>> phi-1:
>>>>>>>
>>>>>>> n^2 + n + phi^2 = 0
>>>>>>>
>>>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>>>
>>>>>>> Keeping things positive,
>>>>>>>
>>>>>>> phi-1:
>>>>>>>
>>>>>>> phi^2 = X
>>>>>>> phi^2 = n'th-root (2^n (L - F^2))
>>>>>>>
>>>>>>> Great, now it looks like I added the Golden Mean among
>>>>>>> the "Factorial/Exponential Identities" given some various
>>>>>>> ideas in laws of large numbers, a square compact Cantor space,
>>>>>>> some expectations in probability, and some sorts these other things.
>>>>>>>
>>>>>>> Borel vs. combinatorics, anyone?
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> Well, whuh, the phi-1 case, ...
>>>>>>
>>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>>>
>>>>>> looks the same as the -phi case, about that,
>>>>>>
>>>>>> -phi:
>>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>> n^2 + n + phi^2 - phi = 0
>>>>>>
>>>>>> phi-1:
>>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>>> n^2 + n + phi^2 - phi = 0
>>>>>>
>>>>>> It looks in both cases that
>>>>>>
>>>>>> phi^2 - phi = X
>>>>>>
>>>>>> X = n'th-root (2^n (L - F^2))
>>>>>>
>>>>>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>
>>>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>>>>>
>>>>>> Because, phi-1 = 1/phi,
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>
>>>>>>
>>>>>> 1/phi:
>>>>>>
>>>>>> n^2 + n + (1/phi)(1/phi+1) = 0
>>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>>>>>> n^2 + n + ((1 +phi)/phi^2) = 0
>>>>>>
>>>>>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>>>>>
>>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>>
>>>>>> -1 -1/phi:
>>>>>>
>>>>>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>>>>>> n^2 + n + 1/phi + 1/phi^2 = 0
>>>>>> 1/phi + 1/phi^2 = 1
>>>>>>
>>>>>>
>>>>>> Hmm, phi: came and went, yielded:
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>
>>>>>> So now I'm looking at the Lucas number,
>>>>>> or that "phi = 1 + 1/phi can be expanded
>>>>>> recursively to obtain a continued fraction
>>>>>> for the golden ratio".
>>>>>>
>>>>>> This is where basically is for getting out
>>>>>> the "infinity'th root" of phi, then to put it in.
>>>>>>
>>>>>> "... one can easily decompose any power of phi into
>>>>>> a multiple of phi and a constant. The multiple
>>>>>> and the constant are always adjacent Fibonacci numbers."
>>>>>>
>>>>>> So I'm looking for the "negative Lucas and Fibonacci
>>>>>> numbers", that phi^n has an expression in L_n and F_n,
>>>>>> looking for phi^{-n}.
>>>>>>
>>>>>> "That pi times root phi is _close_ to 4 is only called
>>>>>> 'numerical coincidence'."
>>>>>>
>>>>>> Let's see, if root phi is "the Golden Root", then,
>>>>>> looking for powers of the Golden Root, ....
>>>>>>
>>>>>>
>>>>>> Hmm, phi: came and went, yielded:
>>>>>>
>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>
>>>>>> Yet, any sort expression n'th powers of phi goes in, ....
>>>>>>
>>>>>> X = phi^2 - phi
>>>>>>
>>>>>> phi^2 - phi - X = 0
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>> The Wiki for phi says an interesting thing, about
>>>>> the consistently small terms in its contined fraction,
>>>>> as explain why the approximates to phi^{-1} converge
>>>>> so slowly: that "this makes the golden ratio an extreme
>>>>> case of the Hurwitz inequality for Diophantine approximations."
>>>>>
>>>>>
>>>>> So looking around some guy says Binet's formulas are like so:
>>>>>
>>>>> Lucas number L_n = phi^n + (-phi)^{-n}
>>>>>
>>>>> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>>>>>
>>>>> Then, I'm looking for phi^{-n}, and n goes to infinity,
>>>>> basically for what are these "negative Lucas and Fibonacci
>>>>> numbers", L_-n and F_-n, and what are these "inverse Lucas
>>>>> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
>>>>> looking for phi^{1/n}.
>>>>>
>>>>>
>>>>>
>>>>> "The ratios between consecutive Fibonacci numbers
>>>>> approaches phi...", in the limit as n -> \infty.
>>>>>
>>>>> I suppose the roots in binary are the same as
>>>>> the roots in "phi-nary". I.e., in their representations.
>>>>>
>>>>> "A root-phi rectangle divides into a pair
>>>>> of Kepler triangles (right triangles with
>>>>> edge lengths in geometric progression)."
>>>>> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>>>>>
>>>>> https://en.wikipedia.org/wiki/Kepler_triangle
>>>>>
>>>>> 1^2 + root-phi ^2 = phi^2
>>>>>
>>>>> root phi = root (phi^2 - 1)
>>>>>
>>>>> "The three edge length 1, root phi, and phi are
>>>>> the harmonic mean, geometric mean, and arithmetic mean,
>>>>> respectively, of the two numbers phi +- 1."
>>>>>
>>>>>
>>>>> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>>>>>
>>>>>
>>>>> About 25/4 and 5/2, ....
>>>>>
>>>>> https://mathworld.wolfram.com/GoldenRatio.html
>>>>>
>>>>> phi = 2 cos (pi/5)
>>>>>
>>>>> Not much to be found about "roots of phi".
>>>>> Or, there's tons to be found "roots of phi" historically,
>>>>> not so much analytically.
>>>>>
>>>>> https://arxiv.org/abs/2312.10767 :
>>>>>
>>>>> "Interestingly, in the non-rotating limit, RFD bifurcates
>>>>> into the usual, non-rotating FD branch and into a spurious
>>>>> branch, named "golden" branch, mapping a non-rotating
>>>>> (static) extremal BH to an under-rotating (stationary)
>>>>> extremal BH, in which the ratio between the angular
>>>>> momentum and the non-rotating entropy is the square
>>>>> root of the golden ratio."
>>>>>
>>>>> https://arxiv.org/abs/2210.13395
>>>>>
>>>>> Found some interest Nigerian papers about
>>>>> "inverse powers of phi", phi^{-n}, but looking
>>>>> for "roots of phi", phi^{1/n}.
>>>>>
>>>>>
>>>>> https://arxiv.org/abs/1512.00379
>>>>>
>>>>> "Psi is the square root of the golden ratio."
>>>>>
>>>>> Lakshmi Roychowdhury, Graf and Luschgy,
>>>>> https://link.springer.com/book/10.1007/BFb0103945
>>>>> still looking for "n'th roots of phi".
>>>>>
>>>>> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>>>>>
>>>>> phi = 2 cos (pi/5)
>>>>> phi/2 = cos(pi/5)
>>>>>
>>>>> nth-root phi/2 = nth-root cos(pi/5)
>>>>>
>>>>> 1/phi = phi - 1
>>>>> phi + 1 - 1 = 2 cos (pi/5)
>>>>> 1/phi + 1 = 2 cos (pi/5)
>>>>> 1/phi = 2 cos (pi/5) - 1
>>>>>
>>>>> phi = 1/ ( 2cos (pi/5) - 1
>>>>> phi/2 = cos (pi/5)
>>>>> arccos (phi/2) = pi/5
>>>>> pi = 5 arccos (phi/2)
>>>>>
>>>>> arccos(-phi/2) = 0
>>>>> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
>>>>> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>>>>>
>>>>> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
>>>>> sin(pi/5) = root (1- (phi/2)^2 )
>>>>>
>>>>> cos^2(theta) = cos(pi/5)
>>>>> sin(theta) = +- root( 1-cos(pi/5) )
>>>>> sin(theta) = += root (1-phi/2)
>>>>> theta = +- arcsin(root(1-phi/2))
>>>>> cos(theta) = root(phi)
>>>>> cos(arcsin(root(1-phi/2))) = root(phi)
>>>>>
>>>>> root phi = 1.27202....
>>>>> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>>>>>
>>>>>
>>>>> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root
>>>>> 5)/2.
>>>>>
>>>>> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
>>>>> Of course that's just modular and doesn't say anything about roots.
>>>>>
>>>>> What I'm looking at is:
>>>>>
>>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>
>>>>> with the idea to write phi^2 - phi in terms of an infinite series,
>>>>> then get all the terms on one side. Then what I figure is that
>>>>> it would be in terms of the n'th root or to-the-1-over-n,
>>>>> getting those on one side, or just the phi part
>>>>>
>>>>> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>
>>>>> ...
>>>>> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>
>>>>> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>>>>>
>>>>> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>> Well I got started figuring out some half-angle formulas,
>>>> with regards to "computing fractional roots of phi"
>>>> with regards to "computing fractional roots of theta"
>>>> with regards to theta being the angle and theta being phi,
>>>> using trigonometry.
>>>>
>>>> Then what I've been looking at is, "infinite series". Euh...,
>>>> infinite series gets involved the "telescoping", and, the
>>>> "scaffolding", with respect to sorts, "reference series",
>>>> in terms of summations and production, "sharing a
>>>> common reference index", that sort of establishing
>>>> telescoping, then, when that collapses down into
>>>> "reference index", for an infinite series, that here
>>>> is prototyped by the inverse powers of two, what
>>>> also happens to be a geometric series, sum of which is 1.
>>>>
>>>> So, I'm looking at such things as deconstructive accounts
>>>> of the geometric series, and a lot that goes on with means,
>>>> with regards to infinite series, reading about Euler and
>>>> this kind of thing, infinite series, with regards to figuring
>>>> out how to help make that the quantities involved,
>>>> result when arithmetic series can or can't have their
>>>> terms grouped and then summed or otherwise
>>>> considered term-wise, infinite series.
>>>>
>>>> I asked around about "do you know any closed forms
>>>> for fractional powers of phi" and it was like "no, and
>>>> I just read a ton of stuff about it", here that what
>>>> I'll be looking at is how to describe the infinite series
>>>> and getting into "surds and adeles" and "the diophantine"
>>>> and "infinite series for e, pi, phi, Catalan's, and so on".
>>>>
>>>>
>>>>
>>>> If you know some particular methodologies about
>>>> infinite series and this kind of thing, I'll be curious
>>>> about establishing what can make for forms, here
>>>> for the "telescoping and scaffolding", what results
>>>> the quantities, result the asymptotics, for various
>>>> law(s) of large numbers, about results like Euler's,
>>>> and Stirling's, then pretty much about continued
>>>> fractions, and infinite series, and addition formula,
>>>> the quantities.
>>>>
>>>>
>>>
>>>
>>>
>>> Continued fractions and phi have something
>>> together where it's [1, 1, 1, 1, ...], about things
>>> like [0, 1, 1, 1, 1, ...], and so on, vis-a-vis infinite
>>> series and things like
>>>
>>> +++++...
>>> +-+-+-+...
>>>
>>> and the idea that "infinite series must have a
>>> point at infinity" then with regards to "also
>>> infinite series oscillate and indicate all the
>>> way out to infinity".
>>>
>>> There's infinite series then there's infinite products,
>>> about all the great utility of "addition formulas"
>>> and getting into "e^x + e^-x" and "sums and differences
>>> of roots and powers and roots and powers of sums
>>> and differences", "addition formulas" and "reciprocity".
>>>
>>> Then this idea of "reference index" and "indexicality",
>>> as well gets into the "infinite metric", about how
>>> the sum of the inverse powers of two equals one,
>>> and very specific references series and products
>>> for the most reduced, as it were, constants, of mathematics.
>>>
>>> https://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan
>>> https://en.wikipedia.org/wiki/Lorenzo_Mascheroni
>>>
>>> https://en.wikipedia.org/wiki/James_Stirling_(mathematician)
>>> https://en.wikipedia.org/wiki/Abraham_de_Moivre
>>>
>>> Submitted, another _belle lettre_ on the stack.
>>>
>>>
>>
>> Today's about n^2 and 2^n.
>>
>> The other day I'm reading some writings of Frege,
>> that were published about fifty years after writing,
>> and he's just mentioning some things while talking
>> about "ideas" and "thoughts", or as might be framed
>> "concepts" and "contingencies", and he wrote:
>>
>> 2^4 = 4^2
>>
>> I looked at this for a second and said to myself,
>> well I'll be danged, that looks like the last one
>> of those in the integers.
>>
>> For a while I've been looking for a natural mathematical
>> constant greater than 4, so I'm wondering about this.
>>
>> 0+0 = 0*0 ? 0^0
>> 1+1 ! 1*1 = 1^1
>> 2+2 = 2*2 = 2^2
>> 3+3 ! 3*3 ! 3^3
>> ....
>>
>> So I'm looking at 2^4 and 4^2,
>> and wondering about x^y = y^x,
>> vis-a-vis xy = yx and x+y = y+x.
>>
>> I sort of recall learning fractional powers in school,
>> and the instructor made a special case for 0^0.
>> I really appreciated this because he made both cases,
>> then explained, that it was a matter of convention
>> and otherwise book-keeping, and as we were quite
>> already in the midst of learning the integral calculus,
>> we had the concepts to consider that it's 0 or 1,
>> which is the convention that it's 1.
>>
>> So now I'm much wondering about x^y = y^x,
>> and about the identities in the forms, or,
>> "where the forms intercept the identity function",
>> here as what's called "the identity dimension".
>>
>> In fractional powers, and 1/oo = 0, and x^0 = 1,
>> gets into "the infinite inverse powers equal 1",
>> another way to look at "roots of unity" than
>> the usual great circle often introduced in the
>> complex analysis, "roots of unity".
>>
>> So, how does this relate to finding forms for
>> fractional powers of phi? It sort of does. The
>> idea is that for any x, there exists y, x =/= y,
>> such that x^y = y^x, or not.
>>
>> 2^4 = 4^2
>>
>> What I'm thinking is that underneath 2, and
>> above two, result two sorts regimes with
>> respect to the exponential, and about how
>> to relate that, with regards to the identity
>> dimension x = y, anywhere that results
>> splitting the quandrant into octants, and
>> having the "surface" their in either octant
>> above and below x = y, then expanding those
>> to each being quadrants, those being interesting
>> new identity dimension surfaces.
>>
>>
>> Then, the idea is that of course exponential
>> grows much, much faster than polynomial,
>> about something it results that there are
>> many, many more roots "inside the box",
>> making for forms that help relate things
>> like finding forms for fractional powers of phi.
>>
>>
>
>
>
> exp <-> pow
>
> 2^3 < 3^2
> 2^4 = 4^2
> 2^5 > 5^2
>
> pow <-> exp
>
> 4^1 > 1^4
> 4^2 = 2^4
> 4^3 < 3^4
>
>
> So, in higher powers in the polynomial,
> and lower radices, bases in the exponent,
> is for for what results
>
> x^y = y^x,
>
> for x or y integer, with regards to here what's called
> "the identity dimension".
>
> So, the identity dimension, is just x = y = z = ...,
> about the usual "x-y plane" with vertical y and
> horizontal x and coordinates as points in R^2
>
> y x=y
> |/_ x
>
> then with the idea that the free variable, of
> the expression, is in the identity dimension, i.d.
>
> The idea is that, "the i.d. only has positive numbers".
>
> Then, as a diagram, it sort of works out for later,
> that the derivations of complex numbers with
> respect to division work out some non-commutative
> complex numbers, a "complex-complex diagram",
> yet here to being it's just the point that basically
> the formulas to draw expressions about the identity
> dimension, are either drawn in the quadrant to show
> how they are above and below it, or, basically in
> the affine about the origin, that the dependent
> variable positive is in octant 2, the dependent
> variable negative is in octant 1, all drawn in
> quadrant 1, then with regards to that angle
> widening, doubling, to draw the result in
> the semi-plane.
>
> Then for example functions symmetric about the
> i.d. x = y, have that the modes of the diagramming,
> make all sorts integral forms and plane curves,
> about these things. (The i.d. sort of is a singularity
> sort of like 0, with regards to things like "the linear
> fractional equation", "Clairaut's equation", "d'Alembert's
> equation", and a bunch of cool integral equations for
> their plane curves, vis-a-vis integral equations and
> their plane curves and the more usual milieu today
> of the differential equations and their systems of
> solutions.)
>
>
> So, the idea with 2^4 and 4^2, is that these are points
> variously about "exp" and "pow", in terms of the
> base or the power being fixed, drawing both those
> either way, where they meet there.
>
> Now why I'm curious about this is because first it's
> that "I know pretty well that 0 is a trivial solution to
> all these differential equations, so this i.d. x = y getting
> into that makes it the envelope for a whole bunch of
> situations in the orbifolds of these integral equations,
> where like 0 it's singular, a singular line as of a singular
> surface about a singular point for multiplicity theory",
> then next because there's this idea of "polynomial and
> exponential the non-polynomial", figuring out "effective
> means to parameterize plane curves of systems of equations
> that meet in the middle to establish completions that
> make for analytical methods".
>
> I started studying differential equations then I started
> studying integral equations because though they are
> each others' derivatives/anti-derivatives, sometimes
> it seems all kind of one-way, and it's not, so the completion
> results build either way, into the middle, middle of nowhere.
>
>
> https://en.wikipedia.org/wiki/Quadrant_(plane_geometry)
>
>
> Then, the idea that especially as it's very graphical
> and results novel diagrams, with utility and for
> geometric applicability, has that this idea of breaking
> down co-ordinates to an "identity dimension" and
> "co-semi-dimensions" and having that all non-negative,
> and non-complex, then gets into using the other quadrants
> for widening the angles variously, or reflections, into
> the negative, and/or the complex, in a novel sort of
> complex-complex diagram on the x-y plane, that usually
> somebody has entirely scribbled over for the Eulerian-Gaussian
> besides all the even and odd the functions as curves with
> respect to symmetry about the origin, helping it out with
> symmetry about this identity dimension in the quadrant,
> thus saving paper and trees and improving the environment.
>
> Of course, it doesn't have the benefit of hundreds of years
> of development on the combined diagram, but think of
> it as an opportunity, to find nice, simple things with neat,
> reproducible visualizability, clear formulaic reformulation,
> and a setting for implicits and the relations of distances
> from the origin and distances down the line, and so on.
>
>
> Here then for the "pow/exp intercepts", is that I noticed
> when somebody mentioned "hey, you know, 2^4 = 4^2",
> while I'm looking phi, the golden ratio,
>
> https://en.wikipedia.org/wiki/Golden_ratio
>
> looking for fractional powers of phi, figuring they
> have diagrams about the fractional powers of integers.
>
>

On 03/05/2024 07:56 AM, Ross Finlayson wrote:
> On 03/04/2024 08:25 PM, Ross Finlayson wrote:
>> On 03/04/2024 10:53 AM, Ross Finlayson wrote:
>>> On 02/24/2024 11:17 AM, Ross Finlayson wrote:
>>>> On 02/24/2024 10:20 AM, Ross Finlayson wrote:
>>>>> On 02/17/2024 09:24 AM, Ross Finlayson wrote:
>>>>>> On 02/16/2024 09:22 PM, Ross Finlayson wrote:
>>>>>>> On 02/16/2024 08:30 PM, Ross Finlayson wrote:
>>>>>>>>
>>>>>>>>
>>>>>>>> So I'm thinking about Factorial/Exponential Identity,
>>>>>>>> from 2003, and one of the outputs is like this.
>>>>>>>>
>>>>>>>> n! ~= root( sum n^n - (sum n) ^n )
>>>>>>>>
>>>>>>>> Now this amuses me a lot, because, it relates these
>>>>>>>> different terms, or rather, in terms of one variable
>>>>>>>> a term, by these various operations, relating the
>>>>>>>> operations together.
>>>>>>>>
>>>>>>>> Here n! is n factorial and root is the square root
>>>>>>>> and sum n^n is sum_i=1^n i^i and (sum n)^n is
>>>>>>>> (n(n+1)/2)^n.
>>>>>>>>
>>>>>>>> So it's funny because as of a root of a difference,
>>>>>>>> it's sort of like a mean
>>>>>>>>
>>>>>>>> arithmetic mean = n'th-part sum-n-terms
>>>>>>>> geometric mean = n'th-root product-n-terms
>>>>>>>>
>>>>>>>> but just about "roots of difference, of various terms".
>>>>>>>>
>>>>>>>> Then, ~= means "approximately equal" it's an approximation,
>>>>>>>> or "it's true in the limit", here leaving that out leaves
>>>>>>>> these kinds of things.
>>>>>>>>
>>>>>>>> F = n!
>>>>>>>> L = sum (n^n)
>>>>>>>> R = (sum n)^n = (n(n+1)/2)^n
>>>>>>>>
>>>>>>>> The most simple sort of algebraic manipulation
>>>>>>>> leads for these kinds of things.
>>>>>>>>
>>>>>>>> F^2 = L - R
>>>>>>>>
>>>>>>>> L = F^2 + R
>>>>>>>>
>>>>>>>> R = L - F^2
>>>>>>>>
>>>>>>>> 2^n (L - F^2) = (n(n+1))^n
>>>>>>>>
>>>>>>>> n'th-root (2^n (L - F^2)) = n^2 + n
>>>>>>>>
>>>>>>>> n'th-root (2^n (L - F^2)) = X
>>>>>>>>
>>>>>>>>
>>>>>>>> n^2 + n - X = 0
>>>>>>>>
>>>>>>>> Now I'm used to seeing something like
>>>>>>>>
>>>>>>>> (n + 1/2)^2 = n^2 + n + 1/4, it's a square
>>>>>>>>
>>>>>>>> which gets me to wondering that
>>>>>>>>
>>>>>>>> X = -1/4
>>>>>>>>
>>>>>>>> when, "n is a root of this quadratic".
>>>>>>>>
>>>>>>>> n'th-root (2^n (L - F^2)) = -1/4
>>>>>>>>
>>>>>>>> (-1/4)^n = 2^n (L - F^2)
>>>>>>>>
>>>>>>>> (-1/8)^n = L - F^2
>>>>>>>>
>>>>>>>> (-1/8)^n = sum (n^n) - n!^2
>>>>>>>>
>>>>>>>> Borel versus Combinatorics, anyone?
>>>>>>>>
>>>>>>>> Fun with Euler? In the limit, ....
>>>>>>>>
>>>>>>>> Yet, for keeping things real, is for something like
>>>>>>>>
>>>>>>>> n^2 + n - X = (n+b)(n-a), where b = a + 1
>>>>>>>>
>>>>>>>> = n^2 + n - ab, or for where
>>>>>>>>
>>>>>>>> X = ab = a(a+1) = a^2 + a
>>>>>>>>
>>>>>>>> when, "n is a root of this quadratic".
>>>>>>>>
>>>>>>>> n'th-root (2^n (L - F^2)) = ab
>>>>>>>>
>>>>>>>> (ab)^n = 2^n (L - F^2)
>>>>>>>>
>>>>>>>> (ab/2)^n = L - F^2
>>>>>>>>
>>>>>>>> (ab/2)^n = sum (n^n) - n!^2
>>>>>>>>
>>>>>>>> ((a^2 + a)/2)^n = sum (n^n) - n!^2
>>>>>>>>
>>>>>>>> Borel versus Combinatorics, anyone?
>>>>>>>>
>>>>>>>> Fun with Euler? In the limit, ....
>>>>>>>>
>>>>>>>> ((a^2) + a)/2) = A
>>>>>>>>
>>>>>>>> A^n = L - F^2
>>>>>>>>
>>>>>>>> A^n/L = 1 - F^2/L
>>>>>>>>
>>>>>>>> Hmm...
>>>>>>>>
>>>>>>>> So, looking for a(a+1) = 1,
>>>>>>>>
>>>>>>>> a^2 + a - 1 = 0
>>>>>>>>
>>>>>>>> I lame out and turn to root finder and inverse symbolic calculator.
>>>>>>>>
>>>>>>>> (for b^2 -4ac/2a, ....)
>>>>>>>>
>>>>>>>> a = (root 5 - 1)/2 ~ 0.618
>>>>>>>> a = (-root5 -1)/2 ~= -1.618
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> https://en.wikipedia.org/wiki/Golden_ratio
>>>>>>>>
>>>>>>>> Hmm, you know they say phi the Golden Mean shows up
>>>>>>>> everywhere, but, the "negative Golden Mean"?
>>>>>>>>
>>>>>>>> Then this, "Golden mean minus one"?
>>>>>>>>
>>>>>>>> So, calling those -phi and phi-1,
>>>>>>>>
>>>>>>>> -phi:
>>>>>>>>
>>>>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>>>>
>>>>>>>> n^2 +n + phi^2 - phi = 0
>>>>>>>>
>>>>>>>> phi-1:
>>>>>>>>
>>>>>>>> n^2 + n + phi^2 = 0
>>>>>>>>
>>>>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>>>>
>>>>>>>> Keeping things positive,
>>>>>>>>
>>>>>>>> phi-1:
>>>>>>>>
>>>>>>>> phi^2 = X
>>>>>>>> phi^2 = n'th-root (2^n (L - F^2))
>>>>>>>>
>>>>>>>> Great, now it looks like I added the Golden Mean among
>>>>>>>> the "Factorial/Exponential Identities" given some various
>>>>>>>> ideas in laws of large numbers, a square compact Cantor space,
>>>>>>>> some expectations in probability, and some sorts these other
>>>>>>>> things.
>>>>>>>>
>>>>>>>> Borel vs. combinatorics, anyone?
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Well, whuh, the phi-1 case, ...
>>>>>>>
>>>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>>>>
>>>>>>> looks the same as the -phi case, about that,
>>>>>>>
>>>>>>> -phi:
>>>>>>> n^2 + n + (-phi)(-phi+1) = 0
>>>>>>> n^2 + n + phi^2 - phi = 0
>>>>>>>
>>>>>>> phi-1:
>>>>>>> n^2 + n + (phi-1)(phi) = 0
>>>>>>> n^2 + n + phi^2 - phi = 0
>>>>>>>
>>>>>>> It looks in both cases that
>>>>>>>
>>>>>>> phi^2 - phi = X
>>>>>>>
>>>>>>> X = n'th-root (2^n (L - F^2))
>>>>>>>
>>>>>>> lim_{n->\infty} (2^n (sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>>
>>>>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi^2 - phi) / 2
>>>>>>>
>>>>>>> Because, phi-1 = 1/phi,
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (phi - 1) phi / 2
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = (1/phi) phi / 2
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>>
>>>>>>>
>>>>>>> 1/phi:
>>>>>>>
>>>>>>> n^2 + n + (1/phi)(1/phi+1) = 0
>>>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>>>> n^2 + n + (1/phi)(1/phi+phi/phi) = 0
>>>>>>> n^2 + n + (1/phi^2 + phi/phi^2) = 0
>>>>>>> n^2 + n + ((1 +phi)/phi^2) = 0
>>>>>>>
>>>>>>> (1+phi)/phi^2 = 1/phi^2 + 1/phi
>>>>>>>
>>>>>>> X = phi^2, when, "n is a root of this quadratic".
>>>>>>>
>>>>>>> -1 -1/phi:
>>>>>>>
>>>>>>> n^2 + n + (-1 -1/phi)(-1/phi) = 0
>>>>>>> n^2 + n + 1/phi + 1/phi^2 = 0
>>>>>>> 1/phi + 1/phi^2 = 1
>>>>>>>
>>>>>>>
>>>>>>> Hmm, phi: came and went, yielded:
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>>
>>>>>>> So now I'm looking at the Lucas number,
>>>>>>> or that "phi = 1 + 1/phi can be expanded
>>>>>>> recursively to obtain a continued fraction
>>>>>>> for the golden ratio".
>>>>>>>
>>>>>>> This is where basically is for getting out
>>>>>>> the "infinity'th root" of phi, then to put it in.
>>>>>>>
>>>>>>> "... one can easily decompose any power of phi into
>>>>>>> a multiple of phi and a constant. The multiple
>>>>>>> and the constant are always adjacent Fibonacci numbers."
>>>>>>>
>>>>>>> So I'm looking for the "negative Lucas and Fibonacci
>>>>>>> numbers", that phi^n has an expression in L_n and F_n,
>>>>>>> looking for phi^{-n}.
>>>>>>>
>>>>>>> "That pi times root phi is _close_ to 4 is only called
>>>>>>> 'numerical coincidence'."
>>>>>>>
>>>>>>> Let's see, if root phi is "the Golden Root", then,
>>>>>>> looking for powers of the Golden Root, ....
>>>>>>>
>>>>>>>
>>>>>>> Hmm, phi: came and went, yielded:
>>>>>>>
>>>>>>> lim_{n->\infty} ( sum (n^n) - n!^2)^{1/n} = 1 / 2
>>>>>>>
>>>>>>> Yet, any sort expression n'th powers of phi goes in, ....
>>>>>>>
>>>>>>> X = phi^2 - phi
>>>>>>>
>>>>>>> phi^2 - phi - X = 0
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> The Wiki for phi says an interesting thing, about
>>>>>> the consistently small terms in its contined fraction,
>>>>>> as explain why the approximates to phi^{-1} converge
>>>>>> so slowly: that "this makes the golden ratio an extreme
>>>>>> case of the Hurwitz inequality for Diophantine approximations."
>>>>>>
>>>>>>
>>>>>> So looking around some guy says Binet's formulas are like so:
>>>>>>
>>>>>> Lucas number L_n = phi^n + (-phi)^{-n}
>>>>>>
>>>>>> Fibonacci number F_n 1 / root 5 (phi^n - (-phi)^{-n}
>>>>>>
>>>>>> Then, I'm looking for phi^{-n}, and n goes to infinity,
>>>>>> basically for what are these "negative Lucas and Fibonacci
>>>>>> numbers", L_-n and F_-n, and what are these "inverse Lucas
>>>>>> and Fibonacci numbers, L_{1/n} and F_{1/n}, really though
>>>>>> looking for phi^{1/n}.
>>>>>>
>>>>>>
>>>>>>
>>>>>> "The ratios between consecutive Fibonacci numbers
>>>>>> approaches phi...", in the limit as n -> \infty.
>>>>>>
>>>>>> I suppose the roots in binary are the same as
>>>>>> the roots in "phi-nary". I.e., in their representations.
>>>>>>
>>>>>> "A root-phi rectangle divides into a pair
>>>>>> of Kepler triangles (right triangles with
>>>>>> edge lengths in geometric progression)."
>>>>>> -- https://en.wikipedia.org/wiki/Dynamic_rectangle
>>>>>>
>>>>>> https://en.wikipedia.org/wiki/Kepler_triangle
>>>>>>
>>>>>> 1^2 + root-phi ^2 = phi^2
>>>>>>
>>>>>> root phi = root (phi^2 - 1)
>>>>>>
>>>>>> "The three edge length 1, root phi, and phi are
>>>>>> the harmonic mean, geometric mean, and arithmetic mean,
>>>>>> respectively, of the two numbers phi +- 1."
>>>>>>
>>>>>>
>>>>>> https://mathshistory.st-andrews.ac.uk/HistTopics/Golden_ratio/
>>>>>>
>>>>>>
>>>>>> About 25/4 and 5/2, ....
>>>>>>
>>>>>> https://mathworld.wolfram.com/GoldenRatio.html
>>>>>>
>>>>>> phi = 2 cos (pi/5)
>>>>>>
>>>>>> Not much to be found about "roots of phi".
>>>>>> Or, there's tons to be found "roots of phi" historically,
>>>>>> not so much analytically.
>>>>>>
>>>>>> https://arxiv.org/abs/2312.10767 :
>>>>>>
>>>>>> "Interestingly, in the non-rotating limit, RFD bifurcates
>>>>>> into the usual, non-rotating FD branch and into a spurious
>>>>>> branch, named "golden" branch, mapping a non-rotating
>>>>>> (static) extremal BH to an under-rotating (stationary)
>>>>>> extremal BH, in which the ratio between the angular
>>>>>> momentum and the non-rotating entropy is the square
>>>>>> root of the golden ratio."
>>>>>>
>>>>>> https://arxiv.org/abs/2210.13395
>>>>>>
>>>>>> Found some interest Nigerian papers about
>>>>>> "inverse powers of phi", phi^{-n}, but looking
>>>>>> for "roots of phi", phi^{1/n}.
>>>>>>
>>>>>>
>>>>>> https://arxiv.org/abs/1512.00379
>>>>>>
>>>>>> "Psi is the square root of the golden ratio."
>>>>>>
>>>>>> Lakshmi Roychowdhury, Graf and Luschgy,
>>>>>> https://link.springer.com/book/10.1007/BFb0103945
>>>>>> still looking for "n'th roots of phi".
>>>>>>
>>>>>> https://mathworld.wolfram.com/TrigonometricPowerFormulas.html
>>>>>>
>>>>>> phi = 2 cos (pi/5)
>>>>>> phi/2 = cos(pi/5)
>>>>>>
>>>>>> nth-root phi/2 = nth-root cos(pi/5)
>>>>>>
>>>>>> 1/phi = phi - 1
>>>>>> phi + 1 - 1 = 2 cos (pi/5)
>>>>>> 1/phi + 1 = 2 cos (pi/5)
>>>>>> 1/phi = 2 cos (pi/5) - 1
>>>>>>
>>>>>> phi = 1/ ( 2cos (pi/5) - 1
>>>>>> phi/2 = cos (pi/5)
>>>>>> arccos (phi/2) = pi/5
>>>>>> pi = 5 arccos (phi/2)
>>>>>>
>>>>>> arccos(-phi/2) = 0
>>>>>> sin(arccos(phi/2) = root ( 1 - (phi/2)^2 )
>>>>>> arccos(phi/2) = arcsin( root ( 1 - (phi/2)^2 )
>>>>>>
>>>>>> pi/5 = arcsin ( root ( 1 - (phi/2)^2 ) )
>>>>>> sin(pi/5) = root (1- (phi/2)^2 )
>>>>>>
>>>>>> cos^2(theta) = cos(pi/5)
>>>>>> sin(theta) = +- root( 1-cos(pi/5) )
>>>>>> sin(theta) = += root (1-phi/2)
>>>>>> theta = +- arcsin(root(1-phi/2))
>>>>>> cos(theta) = root(phi)
>>>>>> cos(arcsin(root(1-phi/2))) = root(phi)
>>>>>>
>>>>>> root phi = 1.27202....
>>>>>> cos(arcsin(square root ( 1 - 2cos(pi/5)))) = ....
>>>>>>
>>>>>>
>>>>>> Some write phi as Phi and 1/Phi as phi. Here though it's (1 + root
>>>>>> 5)/2.
>>>>>>
>>>>>> Hurkens in "Phine Numbers" describes a ring of phine numbers, Z[Phi].
>>>>>> Of course that's just modular and doesn't say anything about roots.
>>>>>>
>>>>>> What I'm looking at is:
>>>>>>
>>>>>> lim_{n->\infty} 2 ( sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>
>>>>>> with the idea to write phi^2 - phi in terms of an infinite series,
>>>>>> then get all the terms on one side. Then what I figure is that
>>>>>> it would be in terms of the n'th root or to-the-1-over-n,
>>>>>> getting those on one side, or just the phi part
>>>>>>
>>>>>> lim_{n->\infty} ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>
>>>>>> ...
>>>>>> ( 2^n sum (n^n) - n!^2)^{1/n} = phi^2 - phi
>>>>>>
>>>>>> 2^n sum (n^n) - n!^2 = (phi^2 - phi)^n
>>>>>>
>>>>>> n!^2 - 2^n sum (n^n) = (phi-phi^2)^n
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>> Well I got started figuring out some half-angle formulas,
>>>>> with regards to "computing fractional roots of phi"
>>>>> with regards to "computing fractional roots of theta"
>>>>> with regards to theta being the angle and theta being phi,
>>>>> using trigonometry.
>>>>>
>>>>> Then what I've been looking at is, "infinite series". Euh...,
>>>>> infinite series gets involved the "telescoping", and, the
>>>>> "scaffolding", with respect to sorts, "reference series",
>>>>> in terms of summations and production, "sharing a
>>>>> common reference index", that sort of establishing
>>>>> telescoping, then, when that collapses down into
>>>>> "reference index", for an infinite series, that here
>>>>> is prototyped by the inverse powers of two, what
>>>>> also happens to be a geometric series, sum of which is 1.
>>>>>
>>>>> So, I'm looking at such things as deconstructive accounts
>>>>> of the geometric series, and a lot that goes on with means,
>>>>> with regards to infinite series, reading about Euler and
>>>>> this kind of thing, infinite series, with regards to figuring
>>>>> out how to help make that the quantities involved,
>>>>> result when arithmetic series can or can't have their
>>>>> terms grouped and then summed or otherwise
>>>>> considered term-wise, infinite series.
>>>>>
>>>>> I asked around about "do you know any closed forms
>>>>> for fractional powers of phi" and it was like "no, and
>>>>> I just read a ton of stuff about it", here that what
>>>>> I'll be looking at is how to describe the infinite series
>>>>> and getting into "surds and adeles" and "the diophantine"
>>>>> and "infinite series for e, pi, phi, Catalan's, and so on".
>>>>>
>>>>>
>>>>>
>>>>> If you know some particular methodologies about
>>>>> infinite series and this kind of thing, I'll be curious
>>>>> about establishing what can make for forms, here
>>>>> for the "telescoping and scaffolding", what results
>>>>> the quantities, result the asymptotics, for various
>>>>> law(s) of large numbers, about results like Euler's,
>>>>> and Stirling's, then pretty much about continued
>>>>> fractions, and infinite series, and addition formula,
>>>>> the quantities.
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>> Continued fractions and phi have something
>>>> together where it's [1, 1, 1, 1, ...], about things
>>>> like [0, 1, 1, 1, 1, ...], and so on, vis-a-vis infinite
>>>> series and things like
>>>>
>>>> +++++...
>>>> +-+-+-+...
>>>>
>>>> and the idea that "infinite series must have a
>>>> point at infinity" then with regards to "also
>>>> infinite series oscillate and indicate all the
>>>> way out to infinity".
>>>>
>>>> There's infinite series then there's infinite products,
>>>> about all the great utility of "addition formulas"
>>>> and getting into "e^x + e^-x" and "sums and differences
>>>> of roots and powers and roots and powers of sums
>>>> and differences", "addition formulas" and "reciprocity".
>>>>
>>>> Then this idea of "reference index" and "indexicality",
>>>> as well gets into the "infinite metric", about how
>>>> the sum of the inverse powers of two equals one,
>>>> and very specific references series and products
>>>> for the most reduced, as it were, constants, of mathematics.
>>>>
>>>> https://en.wikipedia.org/wiki/Eug%C3%A8ne_Charles_Catalan
>>>> https://en.wikipedia.org/wiki/Lorenzo_Mascheroni
>>>>
>>>> https://en.wikipedia.org/wiki/James_Stirling_(mathematician)
>>>> https://en.wikipedia.org/wiki/Abraham_de_Moivre
>>>>
>>>> Submitted, another _belle lettre_ on the stack.
>>>>
>>>>
>>>
>>> Today's about n^2 and 2^n.
>>>
>>> The other day I'm reading some writings of Frege,
>>> that were published about fifty years after writing,
>>> and he's just mentioning some things while talking
>>> about "ideas" and "thoughts", or as might be framed
>>> "concepts" and "contingencies", and he wrote:
>>>
>>> 2^4 = 4^2
>>>
>>> I looked at this for a second and said to myself,
>>> well I'll be danged, that looks like the last one
>>> of those in the integers.
>>>
>>> For a while I've been looking for a natural mathematical
>>> constant greater than 4, so I'm wondering about this.
>>>
>>> 0+0 = 0*0 ? 0^0
>>> 1+1 ! 1*1 = 1^1
>>> 2+2 = 2*2 = 2^2
>>> 3+3 ! 3*3 ! 3^3
>>> ....
>>>
>>> So I'm looking at 2^4 and 4^2,
>>> and wondering about x^y = y^x,
>>> vis-a-vis xy = yx and x+y = y+x.
>>>
>>> I sort of recall learning fractional powers in school,
>>> and the instructor made a special case for 0^0.
>>> I really appreciated this because he made both cases,
>>> then explained, that it was a matter of convention
>>> and otherwise book-keeping, and as we were quite
>>> already in the midst of learning the integral calculus,
>>> we had the concepts to consider that it's 0 or 1,
>>> which is the convention that it's 1.
>>>
>>> So now I'm much wondering about x^y = y^x,
>>> and about the identities in the forms, or,
>>> "where the forms intercept the identity function",
>>> here as what's called "the identity dimension".
>>>
>>> In fractional powers, and 1/oo = 0, and x^0 = 1,
>>> gets into "the infinite inverse powers equal 1",
>>> another way to look at "roots of unity" than
>>> the usual great circle often introduced in the
>>> complex analysis, "roots of unity".
>>>
>>> So, how does this relate to finding forms for
>>> fractional powers of phi? It sort of does. The
>>> idea is that for any x, there exists y, x =/= y,
>>> such that x^y = y^x, or not.
>>>
>>> 2^4 = 4^2
>>>
>>> What I'm thinking is that underneath 2, and
>>> above two, result two sorts regimes with
>>> respect to the exponential, and about how
>>> to relate that, with regards to the identity
>>> dimension x = y, anywhere that results
>>> splitting the quandrant into octants, and
>>> having the "surface" their in either octant
>>> above and below x = y, then expanding those
>>> to each being quadrants, those being interesting
>>> new identity dimension surfaces.
>>>
>>>
>>> Then, the idea is that of course exponential
>>> grows much, much faster than polynomial,
>>> about something it results that there are
>>> many, many more roots "inside the box",
>>> making for forms that help relate things
>>> like finding forms for fractional powers of phi.
>>>
>>>
>>
>>
>>
>> exp <-> pow
>>
>> 2^3 < 3^2
>> 2^4 = 4^2
>> 2^5 > 5^2
>>
>> pow <-> exp
>>
>> 4^1 > 1^4
>> 4^2 = 2^4
>> 4^3 < 3^4
>>
>>
>> So, in higher powers in the polynomial,
>> and lower radices, bases in the exponent,
>> is for for what results
>>
>> x^y = y^x,
>>
>> for x or y integer, with regards to here what's called
>> "the identity dimension".
>>
>> So, the identity dimension, is just x = y = z = ...,
>> about the usual "x-y plane" with vertical y and
>> horizontal x and coordinates as points in R^2
>>
>> y x=y
>> |/_ x
>>
>> then with the idea that the free variable, of
>> the expression, is in the identity dimension, i.d.
>>
>> The idea is that, "the i.d. only has positive numbers".
>>
>> Then, as a diagram, it sort of works out for later,
>> that the derivations of complex numbers with
>> respect to division work out some non-commutative
>> complex numbers, a "complex-complex diagram",
>> yet here to being it's just the point that basically
>> the formulas to draw expressions about the identity
>> dimension, are either drawn in the quadrant to show
>> how they are above and below it, or, basically in
>> the affine about the origin, that the dependent
>> variable positive is in octant 2, the dependent
>> variable negative is in octant 1, all drawn in
>> quadrant 1, then with regards to that angle
>> widening, doubling, to draw the result in
>> the semi-plane.
>>
>> Then for example functions symmetric about the
>> i.d. x = y, have that the modes of the diagramming,
>> make all sorts integral forms and plane curves,
>> about these things. (The i.d. sort of is a singularity
>> sort of like 0, with regards to things like "the linear
>> fractional equation", "Clairaut's equation", "d'Alembert's
>> equation", and a bunch of cool integral equations for
>> their plane curves, vis-a-vis integral equations and
>> their plane curves and the more usual milieu today
>> of the differential equations and their systems of
>> solutions.)
>>
>>
>> So, the idea with 2^4 and 4^2, is that these are points
>> variously about "exp" and "pow", in terms of the
>> base or the power being fixed, drawing both those
>> either way, where they meet there.
>>
>> Now why I'm curious about this is because first it's
>> that "I know pretty well that 0 is a trivial solution to
>> all these differential equations, so this i.d. x = y getting
>> into that makes it the envelope for a whole bunch of
>> situations in the orbifolds of these integral equations,
>> where like 0 it's singular, a singular line as of a singular
>> surface about a singular point for multiplicity theory",
>> then next because there's this idea of "polynomial and
>> exponential the non-polynomial", figuring out "effective
>> means to parameterize plane curves of systems of equations
>> that meet in the middle to establish completions that
>> make for analytical methods".
>>
>> I started studying differential equations then I started
>> studying integral equations because though they are
>> each others' derivatives/anti-derivatives, sometimes
>> it seems all kind of one-way, and it's not, so the completion
>> results build either way, into the middle, middle of nowhere.
>>
>>
>> https://en.wikipedia.org/wiki/Quadrant_(plane_geometry)
>>
>>
>> Then, the idea that especially as it's very graphical
>> and results novel diagrams, with utility and for
>> geometric applicability, has that this idea of breaking
>> down co-ordinates to an "identity dimension" and
>> "co-semi-dimensions" and having that all non-negative,
>> and non-complex, then gets into using the other quadrants
>> for widening the angles variously, or reflections, into
>> the negative, and/or the complex, in a novel sort of
>> complex-complex diagram on the x-y plane, that usually
>> somebody has entirely scribbled over for the Eulerian-Gaussian
>> besides all the even and odd the functions as curves with
>> respect to symmetry about the origin, helping it out with
>> symmetry about this identity dimension in the quadrant,
>> thus saving paper and trees and improving the environment.
>>
>> Of course, it doesn't have the benefit of hundreds of years
>> of development on the combined diagram, but think of
>> it as an opportunity, to find nice, simple things with neat,
>> reproducible visualizability, clear formulaic reformulation,
>> and a setting for implicits and the relations of distances
>> from the origin and distances down the line, and so on.
>>
>>
>> Here then for the "pow/exp intercepts", is that I noticed
>> when somebody mentioned "hey, you know, 2^4 = 4^2",
>> while I'm looking phi, the golden ratio,
>>
>> https://en.wikipedia.org/wiki/Golden_ratio
>>
>> looking for fractional powers of phi, figuring they
>> have diagrams about the fractional powers of integers.
>>
>>
>
>
> https://www.youtube.com/watch?v=yfKD9NrFPBA&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=10
>
>
> https://en.wikipedia.org/wiki/Singular_point_of_a_curve
> https://en.wikipedia.org/wiki/Singular_solution
> https://en.wikipedia.org/wiki/Envelope_(mathematics)
> https://en.wikipedia.org/wiki/Subtangent
>
> https://en.wikipedia.org/wiki/Linear_fractional_transformation
> https://en.wikipedia.org/wiki/Clairaut%27s_equation
> https://en.wikipedia.org/wiki/D%27Alembert%27s_equation
>
> https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
> https://en.wikipedia.org/wiki/Peano_existence_theorem
> https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem
>
> https://en.wikipedia.org/wiki/Identity_function
> https://en.wikipedia.org/wiki/Origin_(mathematics)
> https://en.wikipedia.org/wiki/Domain_(mathematical_analysis)
> https://en.wikipedia.org/wiki/Interval_(mathematics)
>
> https://en.wikipedia.org/wiki/Donald_Sarason
> https://en.wikipedia.org/wiki/Constantin_Carath%C3%A9odory
>
> https://en.wikipedia.org/wiki/Integral_equation
>
> https://en.wikipedia.org/wiki/Numerical_continuation
> https://de.wikipedia.org/wiki/Umlaufsatz
>
>
> That video then gets into five or six hours of
> consideration of "identity dimension" analysis,
> and a lot of it is about the singular, which of
> course is just a branch, in a multiplicity.
>
> One idea introduced here is "further roots
> of unity" or "factors of unity", as was considered
> in the above "roots of zero", about that x^0 = 1.
>
> Then, here mostly the idea that the integral equations
> and differential equations are fundamentally reciprocal,
> and that interesting plane curves like x = y are envelopes
> of integral equations, and about things like Fourier-style,
> where an infinite sum of non-linear functions results
> linear functions, vis-a-vis, usually enough the other way
> around, helps explore the dynamics in definition, that
> get involved, for a unified sort of approach to the
> differential equations as the integral equations,
> and then for above about the existence results,
> and uniqueness results, that there's a lot going on
> besides the cool way we have today, and when
> definitions like "curve" and "differentiable" get
> involved, that there's a wider world's definitions,
> vis-a-vis classical functions, Cartesian functions,
> curves, tangent and turning tangent, and a bunch
> other notions to sort out, helping de-mystify this
> sort of original analysis, toward a thorough treatment
> of the field.
>