For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime. Denote by $PPG_{a,b}(m)$ the $m$-th prime power gap of type $(a,b)$ that is, twice the $m$-th smallest Galois radius of type $(a,b)$ of the smallest possible integer $u$.
Can we get lower and/or upper bounds on $PPG_{a,b}(m)$ in terms of $a$, $b$ and $m$ both unconditionally and under widely believed conjectures such as (G)RH, Elliott-Halberstam conjecture or Hardy-Littlewood $k$-tuple conjecture?
