Comments on: Powers of Phi

By: John Boyer

John Boyer — Sat, 14 Sep 2024 02:31:22 +0000

There’s an even easier way to find the nth power of phi. I found it today.

phi^n = (phi)f(n) + f(n-1), where f(n) is the nth fibonacci number.

Once I found it, it was pretty easy to prove by induction.

And it seems that (1/phi)^n = |f(n-1) – (1/phi)f(n)|, but I haven’t tried proving that one yet.

By: petergrfstrm

petergrfstrm — Wed, 28 Aug 2024 12:07:46 +0000

phi^înteger is approaching an integer exponentially as n increases
it is a so called Pisot–Vijayaraghavan number
https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number

n phi^n epsilon
34 12 752 043 -7,84e-8
47 6 643 838 879 1,505e-10

By: Reid B

Reid B — Mon, 15 Jan 2024 01:00:06 +0000

In reply to Gary B Meisner. Sqrt(2) + 1 also has this property, as does (3+sqrt(13))/2. The thing I found in common with all of these numbers is that their continued fractions are all uniform, with the same number repeating infinitely.

By: Wasicki

Wasicki — Mon, 06 Nov 2023 03:02:43 +0000

Powers of phi can also be related to Lucas numbers through an infinite continued fraction:

phi^n = L(n) – (-1)^n/(L(n) – (-1)^n/(L(n) – (-1)^n/(L(n) – … )))

For example,

phi^3 = 4 + 1/(4 + 1/(4 + 1/(4 + … ))) = 4.236067977499789696409173668731…

By: Geoff Petrasek

Geoff Petrasek — Wed, 15 Feb 2023 18:41:15 +0000

In reply to Heyhey1D. Phi is found in the power of the sun. It represents natures response to energy which is to maximize its use thru genetic adaptation. For example, the spacing of leaves, and seeds to maximize growth. Phi is a sacred gift from God.

By: Neal

Neal — Thu, 15 Dec 2022 21:09:19 +0000

Playing around in MS Mathematics I found that

Phi^(n) + Phi^(n+1) = Phi^(n+2)

Is just one instance of the more general

Phi^(n/a) + Phi^((n+a)/a) = Phi^((n+2a)/a)
Where a does not equal 0 because zero root is undefined.

The top example is a=1.

I’ve searched for other mentions of this online, but could not find any. My searches have led me here.

By: Ramsey

Ramsey — Wed, 05 Oct 2022 15:32:09 +0000

where multiplication and addition of exponents of Phi are the same, which is a marveollous thing.

By: Secret

*Secret* — Mon, 14 Jun 2021 12:14:35 +0000

The second equation is right

By: Dav Jons

Dav Jons — Mon, 12 Apr 2021 09:17:34 +0000

Oh! That is an interesting topic I also have a query about this topic. In this way, the ratio of terms of the Fibonacci sequence approaches phi. The sequence of the element's approach (phi^n)/sqrt(5) on the way of sequence approaches a geometric. One could take the general formula for the sum of a geometric series to at least approximate the sums you began all, but it’d take an impartially large 'n' to get good results. I just need to found this type of blog and also gives a lot of help in such solve fractive issue , and I’m really enjoying it!.

By: M.D.

M.D. — Sun, 20 Sep 2020 16:13:54 +0000

Roots of Lucas numbers tend towards Φ
199^(1/10)=1,61803….