<s>[INST] To solve $1+2^mp^2=q^5$ How do we find all posible solutions of $1+2^mp^2=q^5$ for positive integer $m$ and primes $p,q$ ? $m=1,q=3,p=11$ is a solution , is there any other solution ?
 [/INST] Following Ivan Loh, $2^mp^2=(q-1)(q^4+q^3+q^2+q+1)$ implies $(q^4+q^3+q^2+q+1)$ equals either $p$ or $p^2$ (because it's odd).  In either case $p\gt q^2$, which implies $p$ cannot divide $q-1$, hence we must have $q^4+q^3+q^2+q+1=p^2$ and thus $q-1=2^m$.  Writing now $q=2^m+1$, we wind up with
$$2^{4m}+5\cdot2^{3m}+5\cdot2^{2m+1}+5\cdot2^{m+1}+5=p^2$$
Because $p^2\equiv1$ mod $8$ for any odd number, we must have $m\lt2$.  This leaves only $m=1$, $q=3$, $p=11$.
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