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
]

Generate a big list of Fibonacci numbers:

fibList=fibonacciList[10000];

fibList[[1]]

Adjust the indexes to match the usual mathematical notation:

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

Try it out:

f[0]

f[1]

f[10]

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?

Compute (Subscript[F, n+1])(Subscript[F, n-1]) for various n:

list1=Table[f[n+1]f[n-1],{n,1,20}]

Compute ((Subscript[F, n])^2) for various n:

list2=Table[f[n]^2,{n,1,20}]

TableForm[Transpose[{list1,list2,list1-list2}],TableHeadings->{None,{"(Subscript[F, n + 1])(Subscript[F, n - 1])","((Subscript[F, n])^2)","difference"}}]

Observation: (Subscript[F, n+1])(Subscript[F, n-1])-((Subscript[F, n])^2)=(((-1))^n)

Does this hold for larger n?

list1=Table[f[n+1]f[n-1]-f[n]^2,{n,1001,1020}]

A conjecture is a precise mathematical statement that we believe to be true based on some observations, but we haven’t proved.

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

Generalize: What happens if we consider  (Subscript[F, n+2])(Subscript[F, n-2])-((Subscript[F, n])^2)? Or (Subscript[F, n+3])(Subscript[F, n-3])-((Subscript[F, n])^2)? Or (Subscript[F, n+4])(Subscript[F, n-4])-((Subscript[F, n])^2)? Etc.

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

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

testConjecture[i_]:=Table[{f[n+i]f[n-i]-f[n]^2,(-1)^(n+i-1)*f[i]^2},{n,i,50}]

testConjecture[1] (* i=1 *)

testConjecture[2] (* i=2 *)

testConjecture[3] (* i=3 *)

testConjecture[4] (* i=4 *)

testConjecture[8] (*i=8*)

testConjecture[20] (*i=20*)