How to print intermediate steps of simplifying a power formula?

Question

To answer the question of proving Fibonacci sequence is periodic mod 5 without using induction., I came across Mathematica to prove $$F_{n}\equiv F_{n+20}\pmod 5$$ for all $n \geq 2$

I defined:

$F[n] := \cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^n-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^n$

and showed:

Simplify[f[n + 20] - f[n] - 10945 f[n] - 6765 f[n - 1]]

equals to 0. (I knew the fact that $F_{n+20} = 10946 F_{n}+ 6765 F_{n-1}$, but I wanted to reach a proof without induction)

Thus, for all $n \geq 2$ we have $$F_{n}\equiv F_{n+20}\pmod 5$$

The problem is that I can not get the intermediate steps of Simplify function, even using the Trace commands. Could you please help?

If possible, I want to also show this in Wolfram Alpha. I don't know how to run several commands in Wolfram Alpha.

Fibonacci[] is built-in. Also, Simplify[] was never meant to show intermediate steps, as what it does under the hood is rather different from what a human would do. — J. M.'s missing motivation, Jul 13 '15 at 18:03
@Guesswhoitis. If you read my answer at math.SE, you see that I didn't want to calculate Fibonacci[n] with a reursive formula defined by induction. Because of this, I have used the closed form of Binet's formula. https://en.wikipedia.org/wiki/Jacques_Philippe_Marie_Binet — Ho1, Jul 13 '15 at 18:09
Yes, and FunctionExpand[Fibonacci[n]] will give you (a continued version of) Binet's formula. Whatever gave you the idea that Fibonacci[] is unsuitable here? (FWIW, I am somewhat familiar with Binet.) — J. M.'s missing motivation, Jul 13 '15 at 18:11
Do you have any idea about how can I get the intermediate steps of simplifying? — Ho1, Jul 13 '15 at 18:13
Again: Mathematica does not work like that. It doesn't do things the way a human might, so not much point in having it "show its work". — J. M.'s missing motivation, Jul 13 '15 at 18:15
Do you mean that there is a possiblity that Mathematica does this simplifying numerically? It outputs exact Zero. — Ho1, Jul 13 '15 at 18:18
I am merely saying that a computer algebra system will in general not manipulate symbols in the same way a human might, nothing more. — J. M.'s missing motivation, Jul 13 '15 at 18:20
So, I can say that my proof is mathematically correct. But I have a trouble to show others the proof. :-( Anyway, there should be a way, so I can see that how it works, no matter how different it does this from a human. — Ho1, Jul 13 '15 at 18:23
"The computer is a precision tool. It should not be used as a bludgeon or a substitute for thought." - F. S. Acton (See this.) — J. M.'s missing motivation, Jul 13 '15 at 18:28
I don't want it to think instead of me, I want it to show the intermediate steps of simplifying an algebraic expression. — Ho1, Jul 13 '15 at 19:05

How to print intermediate steps of simplifying a power formula?

0 Answers0