October 21, 2009
The
expression $a|b$ means
that $a$ divides $b$. Suppose that $2^{m}|(3^{m}-1)$.
Show
that if $m\ne 1$, then $m$ is even.
Solution.
If $m$ is odd, then $3^m \equiv (-1)^m \equiv -1 \pmod {4}$, so
$3^m-1 \equiv 2 \pmod {4}$ and is not divisible by $2^m$ for odd $m$
larger than 1.