Primality Criteria for Pairs $n$ and $n+d$

Abstract

The existence of primality criteria for generic pairs $n$ and $n+d$ is investigated. A congruence $\pmod {n(n+d)}$ is found, that holds if and only if $(n,n+d)$ is a prime pair, except for a finite number of exceptions that appear when $n$ is lower than a fixed quantity only depending on $d$. Explicit primality criteria for $d = 2,4,6,8,10,12$ are given and a formula predicting the number of exceptions is conjectured.