$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$ Here $\varphi = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?
Prove: the intersection of Fibonacci sequence and Mersenne sequence is just $\{1,3\}$
8
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sequences-and-series
elementary-number-theory
fibonacci-numbers
1 Answers
11
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.
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0@Douglas: Ah, of course, that makes sense! Neat argument. (The statement does remain true with 9 replaced by 8, though, which was what confused me.) – 2011-01-07