Tight bound for the density of sequence of integers the sum of no two of which is a perfect square

Abstract

Erdős and Sárkőzy proposed the problem of determining the maximal density attainable by a set S of positive integers having the property that no two distinct elements of S sum up to a perfect square. Massias [(Sur les suites dont les sommes des terms 2 á 2 ne sont par des carr)] exhibited such a set consisting of all x≡1 ( mod 4) with x≡14,26,30 ( mod 32) . Lagarias et al. [(J. Combin. Theory Ser. A 33 (1982) 167)] showed 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 essentially proving that the Massias’ set has the best possible density. They [(J. Combin. Theory Ser. A 34 (1983) 123)] also proved that the density of such a set S is never >0.475 when we allow general sequences. We improve on the lower bound for general sequences, essentially proving that it is not 0.475, but arbitrarily close to 11 32 , the same as that for sequences made up of only arithmetic progressions.