On the density of sequences of integers the sum of no two of which is a square. I. Arithmetic progressions

Abstract

The maximal density attainable by a sequence S of positive integers having the property that the sum of any two elements of S is never a square is studied. J. P. Massias exhibited such a sequence with density 11 32 ; it consists of 11 residue classes (mod 32) such that the sum of any two such residue classes is not congruent to a square (mod 32). It is shown that for any positive integer n, one cannot find more than 11 32 n residue classes (mod n) such that the sum of any two is never congruent to a square (mod n). Thus Massias' example has maximal density among those sequences S made up of a finite set of (infinite) arithmetic progressions. A companion paper will bound the maximal density of an arbitrary such sequence S.