Say $r$ is an Asperger radius of $n$ if $(n-r,n+r)=(p^{a},q^{b})$ with $p$ and $q$ primes and integral $a$ and $b$ such that $ab=r$. If $r$ is a divisor of a prime number then of course $\min(a,b)=1$.
The famous twin prime conjecture can be reformulated as : $1$ is an Asperger radius of infinitely many integers.
Are there infinitely many integers admitting at least one Asperger radius?