[Edu-sig] Fibonacci Numbers and Phi (again)

[kirby urner]

> On Fri, Nov 22, 2013 at 9:49 PM, Litvin <litvin at skylit.com> wrote:

> Kirby,
>
> I am sorry to spoil all the fun, but this is a not a gem but a
> mathematical trick, and I think the way to deal with
> mathematical tricks is to explain them, not to verify with code.


Having an algebraic verification is more like the proof I did not provide.
Thanks for the reasoning Gary.

I do think there's a two phase understanding for theorems where phase one
is too oft neglected, which is understanding what the theorem is claiming
or asserting, before attempting an analysis of "why?".

Euler's Theorem by way of example, the one with the totient, a
generalization of Fermat's:  it's too easy to just stare at the assertion
and not think of examples, not know what it says.

This is where I think a Python program may come in handy, not as a proof of
a theorem but as a illustration or demonstration of its consequences.  See
the theorem in action somehow.

Seeing something in Python that's runnable may close some critical circuits
in the brain of the theorem's reader.  "Now I better know what it means"
may be the aha experience.

Most people who have read up on Fibonaccis know they can be extended in the
negative direction.  Then the idea of starting with any two numbers, not
necessarily integers may be introduced.

The limit of F[N+1]/F[N] and its relationship to PHI (equal as N ->
infinity) may then be discussed.

I've taken exactly this same approach with one of Ramanujan's for 1/pi
(just flip it for pi).  A little Python generator will do the trick.  In
this case the gem being verified has not to my knowledge been fully
explained or reasoned about (the proof has yet to be supplied):

http://worldgame.blogspot.com/2012/01/testing-math-ml.html

(this blogpost uses MathJax, a good resource for the JavaScript-enabled)

I'm still without an explanation for this pi digits generator from one of
Michel Paul's students, but I share it around in hopes of finding one --
and then I hope its one I can follow.

https://groups.google.com/forum/#!msg/mathfuture/LA0pMPC6-HE/MBGWxn4ENsUJ

Thanks again for clearing up any mysteries that may have attached to the
PHI ** N formula (I originally called it a formula, not a gem... some gems
are only semi-precious).

Kirby
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