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Tight bound for the density of sequence of integers the sum of no two of which is a perfect square. (English) Zbl 1066.11004
Summary: P. Erdős and A. Sárközy [Bull. Greek Math. Soc. 18, 204–223 (1977; Zbl 0413.10049), p. 209] 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. J.-P. Massias [Sur les suites dont les sommes des terms 2 à 2 ne sont pars des carrés. Publications du Département de mathématiques de Limoges (1982)] exhibited such a set consisting of all $$x\equiv 1 \pmod 4$$ with $$x\equiv14,26,30 \pmod {32}$$. J. C. Lagarias et al. [J. Combin. Theory, Ser. A 33, 167–185 (1982; Zbl 0489.10052)] showed that for any positive integer $$n$$, one cannot find more than $$11n/25$$ 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, 123–129 (1983; Zbl 0514.10041)] 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 Image, the same as that for sequences made up of only arithmetic progressions.

##### MSC:
 11B05 Density, gaps, topology 11B75 Other combinatorial number theory 11L03 Trigonometric and exponential sums, general
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