On the density of sequences of integers the sum of no two of which is a square II. General sequences

J. C. Lagarias; A. M. Odlyzko; J. B. Shearer

doi:10.1016/0097-3165(83)90051-1

On the density of sequences of integers the sum of no two of which is a square II. General sequences

J. C. Lagarias
, A. M. Odlyzko
, J. B. Shearer

School of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N₀ such that for all N ≥ N₀ any set S ⊆ [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.

Original language	English (US)
Pages (from-to)	123-139
Number of pages	17
Journal	Journal of Combinatorial Theory, Series A
Volume	34
Issue number	2
DOIs	https://doi.org/10.1016/0097-3165(83)90051-1
State	Published - Mar 1983

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abstract = "The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N0 such that for all N ≥ N0 any set S ⊆ [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.",

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N2 - The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N0 such that for all N ≥ N0 any set S ⊆ [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.

AB - The aim of this paper is to study the maximal density attainable by a sequence S of positive integers having the property that the sum of any two distinct elements of S is never a square. It is shown that there is a constant N0 such that for all N ≥ N0 any set S ⊆ [1, N] having this property must have |S| < 0.475N. The proof uses the Hardy-Littlewood circle method.

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On the density of sequences of integers the sum of no two of which is a square II. General sequences

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