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.