Simultaneous primality of the integers $n$ and $2n - d$

Abstract

A necessary and sufficient condition for the simultaneous primality of integers $n$ and $2n-d$ is given by means of congruences ${mod n(2n - d)}$ that hold if and only if they form a prime pair. These are used to obtain explicit primality criteria for some values of $d$, after computation of a finite number of exceptions that appear when $n$ is lower than a fixed quantity depending only on $d$.