I was considering some number theory problems which inspired me to write the following conjecture, which bears some resemblance to the Catalan problem, but is in fact different:
Fix two distinct sequences of primes $p_{1}, ..., p_{n}$ and $q_{1}, ..., q_{m}$. Do there exist infinitely many sequences of naturals $a_{1}, ..., a_{n}$, $b_{1}, ..., b_{m}$ such that:
$p_{1}^{a_{1}} ... p_{n}^{a_{n}} - q_{1}^{b_{1}} ... q_{m}^{b_{m}} = 1$?