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

Andrew Bremner, Tho Nguyen Xuan

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Bremner, Guy and Nowakowski [Which integers are representable as the product of the sum of three integers with the sum of their reciprocals? Math. Compos. 61(203) (1993) 117-130] investigated the Diophantine problem of representing integers n in the form (x + y + z)(1/x + 1/y + 1/z) for rationals x,y,z. For fixed n, 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 n, and remarked that it seemed plausible that there were always solutions with positive w,x,y,z when n ≥ 16. This is false, and the situation is quite subtle. We show that there cannot exist such positive solutions when n is of the form 4m2, 4m2 + 4, where m ≢ 2 (mod 4). Computations within our range seem to indicate that solutions exist for all other values of n.

Original languageEnglish (US)
Pages (from-to)1229-1246
Number of pages18
JournalInternational Journal of Number Theory
Issue number5
StatePublished - Jun 1 2018


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

ASJC Scopus subject areas

  • Algebra and Number Theory


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