This question is a weaker version of the twin prime problem.
Question: Are there infinitely many prime powers $p^a, q^b$ with $p^a-q^b=2$?
Of course, we expect an answer easier than solving the twin prime problem.
The examples for $p,q<20000$ and $a,b<10$ with $(a,b) \neq (1,1)$ are the following:
$3^3- 5^2$,
$3^2-7$,
$3^4-79$,
$3^5-241$,
$3^6-727$,
$3^9-19681$,
$5^2-23$,
$7^2-47$,
$7^4-2399$,
$11-3^2$,
$11^4-14639$,
$13^2-167$,
$19^2-359$,
$19^3-6857$,
$29-3^3$,
$29^2-839$,
$37^2-1367$,
$43^2-1847$,
$47^2-2207$,
$61^2-3719$,
$71^2-5039$,
$83-3^4$,
$89^2-7919$,
$103^2-10607$,
$107^2-11447$,
$127-5^3$,
$127^2-16127$,
$131^2-17159$,
$139^2-19319$,
$6563-3^8$
This list leads to the following question: Is there an example with $1 \not \in \{a,b\}$ and $(p,q) \neq (3,5)$?