### Abstract

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.

Original language | English |
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Pages (from-to) | 243-255 |

Number of pages | 13 |

Journal | Discrete Mathematics |

Volume | 256 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Sep 28 2002 |

### Keywords

- Additive number theory
- Combinatorial number theory

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*256*(1-2), 243-255. https://doi.org/10.1016/S0012-365X(01)00435-6