Erdos and Sárkozy 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 11/32n 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.
- Additive number theory
- Combinatorial number theory
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics