"On the Density of Sequences of Integers the Sum of no Two of Which is a Square I. Arithmetic Progressions" with J.C. Lagarias and A.M. Odlyzko, Journal of Combinatorial Theory Series A, 33(1982), p. 167-185.

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.

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