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$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$ Here $\varphi = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?

1 Answers 1

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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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    @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