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A diophantine problem concerning cubes

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
110
Issue
2
Identifiers
DOI: 10.1016/j.jnt.2004.07.013
Keywords
  • Pairwise Sums Of Integers
  • Equal Sums Of Cubes
  • Diophantine Chains

Abstract

Abstract This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2 ( n + 1 ) integers, in parametric terms, such that 5 n + 1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5 n + 2 of the pairwise sums being cubes.

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