Fibonacci Identities

Math 242 Modern Computational Mathematics

Preliminary Code

Here is one of our fibonacciList modules from last class:

fibonacciList[n_]:=Module[{fibList=Table[0,n+1]},
(* set the initial values (Subscript[F, 0]) and (Subscript[F, 1]) *)
fibList[[2]]=1;

(* compute (Subscript[F, 2]) through (Subscript[F, n]) and store them in the list *)
Do[
fibList[[i]]=fibList[[i-1]]+fibList[[i-2]]
,{i,3,n+1}];

(* return the list *)
fibList
]

First, generate a big list of Fibonacci numbers and store them in memory.

fibList=fibonacciList[10000];

fibList[[1]]

Create a little function to adjust the indexes to match the standard mathematical notation.

f[n_]:=fibList[[n+1]]

f[0]

f[1]

f/@Range[0,10]

Last time, we computed (Subscript[F, n+1])(Subscript[F, n-1]) and ((Subscript[F, n])^2) for various n. What did we observe?

Let's make a list of values (Subscript[F, n+1])(Subscript[F, n-1]):

Table[f[n+1]*f[n-1],{n,1,20}]

Table[f[n]^2,{n,1,20}]

Table[f[n+1]*f[n-1]-f[n]^2,{n,1,20}]

It looks like (Subscript[F, n+1])(Subscript[F, n-1])-((Subscript[F, n])^2)=(((-1))^n). Does this hold for larger n?

Table[f[n+1]*f[n-1]-f[n]^2,{n,101,110}]

Conjecture: The Fibonacci numbers satisfy (Subscript[F, n+1])(Subscript[F, n-1])-((Subscript[F, n])^2)=(((-1))^n) for every integer n>=1.

A conjecture is a precise statement of an unproven mathematical idea that we think is true.

Generalize: What if we replace n+1 with n+2, and n-1 with n-2? How about n-3 and n+3? Etc.

Conjecture: The Fibonacci numbers satisfy (Subscript[F, n+k])(Subscript[F, n-k])-((Subscript[F, n])^2)=(((-1))^(n+k-1))((Subscript[F, k])^2) for positive integers n and k.

Table[{f[n+k]*f[n-k]-f[n]^2,(-1)^(n+k-1)*f[k]^2},{n,1,10},{k,1,n}]

testConjecture[k_]:=Table[{f[n+k]*f[n-k]-f[n]^2,(-1)^(n+k-1)*f[k]^2},{n,k,10}]

k = 1 :

testConjecture[1]

k = 2 :

testConjecture[2]

k = 3 :

testConjecture[3]

k = 4 :

testConjecture[4]