Kirby,
You are right, a little program does help to clarify the statement
and the formulas involved. But then, keep it really simple (no
classes, no generators), so that understanding of the code doesn't
get in the way. For example:
from math import sqrt
rt5 = sqrt(5)
phi = (1 + rt5)/2
def neat_formula(f0, f1, f2):
return (f0 + f2 + rt5*f1)/2
power_of_phi = phi**(-6)
f0, f1, f2 = 13, -8, 5
for k in range(20):
print("{0:8.5f} {1:8.5f}".format(power_of_phi, neat_formula(f0, f1, f2)))
f0, f1, f2 = f1, f2, f1+f2
power_of_phi *= phi
There are many other occasions to bring in more advanced tools.
Gary Litvin
www.skylit.com
At 11:08 AM 11/23/2013, kirby urner wrote:
> On Fri, Nov 22, 2013 at 9:49 PM, Litvin
<<mailto:lit...@skylit.com>lit...@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>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>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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