The equation (w + x + y + z)(1/w + 1/x + 1/y + 1/z) = n

Andrew Bremner, Tho Nguyen Xuan

Research output: Contribution to journalArticle

Abstract

Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Comp. 61(203) (1993) 117–130] investigated the Diophantine problem of representing integers (Formula presented.) in the form (Formula presented.) for rationals (Formula presented.). For fixed (Formula presented.), the equation represents an elliptic curve, and the existence of solutions depends upon the rank of the curve being positive. They observed that the corresponding equation in four variables, the title equation here (representing a surface), has infinitely many solutions for each (Formula presented.), and remarked that it seemed plausible that there were always solutions with positive (Formula presented.) when (Formula presented.). This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when (Formula presented.) is of the form (Formula presented.), (Formula presented.), where (Formula presented.). Computations within our range seem to indicate that solutions exist for all other values of (Formula presented.).

Original languageEnglish (US)
Pages (from-to)1-18
Number of pages18
JournalInternational Journal of Number Theory
DOIs
StateAccepted/In press - Jan 25 2018

Keywords

  • Diophantine representation
  • Elliptic curve
  • Hilbert symbol
  • quartic surface

ASJC Scopus subject areas

  • Algebra and Number Theory

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