Question: Is there a solution of $p^a-q^b = 2$ with $p,q$ prime and $a,b > 1$, except $3^3-5^2$ ?
Remark: For $p,q,a,b < 200$, the only solution is $3^3-5^2$.
Idem for $\max(p,q)<200000$, $\min(p,q)<200$ and $a,b<20$.
This question came after this post: Infiniteness of twin prime powers
It is strongly suspected that the only solutions in positive integers $x, y, n$ and $m$ with $n, m \geq 2$ to $$ x^n-y^m=2 $$ is with $(x,y,n,m)=(3,5,3,2)$. Proving this (even in the special case with $x$ and $y$ prime) is apparently beyond current technology (unless one has a proof of the $ABC$-Conjecture). Mihailescu's proof of Catalan's Conjecture relies fundamentally upon properties of cyclotomic fields and does not extend to this case.