TITLE: Prove: the intersection of Fibonacci sequence and Mersenne sequence is just $\{1,3\}$
QUESTION [8 upvotes]: $$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$$
Here $\varphi  = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?

REPLY [11 votes]: We need to find when $F_n+1$ is a power of 2.  Almost every value of $n$ can be eliminated by considering the Pisano period.  In particular, we can deduce that:

$F_n+1 \equiv 0 \pmod {16}$ if and only if $n \equiv 22 \pmod {24}$ and
$F_n+1 \equiv 0 \pmod 9$ if $n \equiv 22 \pmod {24}$.

This leaves the few small cases already listed.