Assume that the sequence of Fibonacci numbers begins with 2 instead of 1,i.e., $$F_1=2,F_2=3,F_3=5,...$$
Let $F_n$ denote the $n^{th}$ Fibonacci number. I have to prove these two claims.
Claim 1: $F^2_{n+1} = 1 \mod F_n$.
Claim 2: $F^2_{n} = -1 \mod F_{n+1}$.
I do not know where to go with this. I have probably missed something that obvious in the "induction" step of proof. But I cannot realize it.
Any help or suggestion?
You have $F_n^2-F_{n+1}F_{n-1}=(-1)^{n+1}$ which you should be able to prove, and then use.
With these kind of things it is sometimes worth looking for a stronger statement which is more amenable to the induction you are trying to use.