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

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 32n 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.