A diophantine problem concerning cubes

✍ Scribed by Ajai Choudhry

• Book ID: 104024490
• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 188 KB
• Volume: 110
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.07.013

✦ Synopsis

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 5n + 1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n + 2 of the pairwise sums being cubes.