Searching for Solutions of x3 + y3 + z3 = k

  • D. R. Heath-Brown
Part of the Progress in Mathematics book series (PM, volume 102)


Diophantine equations of the form

in which k is a given positive integer, and the unknowns x, y, z can be any integers, positive, negative or zero, have been studied by a number of authors. In particular it has been asked whether there are any solutions for k = 3 other than (x,y, z) = (1,1,1) or (4, 4, –5); and whether there are any solutions at all for k = 30. Computer investigations by Gardiner, Lazarus and Stein [1] in 1964 failed to resolve these questions.


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  1. [1]
    V.L. Gardiner, R.B. Lvarus and P.R. Stein.–Solutions of the Diophantine equation æ3 + y3 = z3 — d, Math. Comp. 18, (1964), 408–413.MathSciNetzbMATHGoogle Scholar
  2. [2]
    W.J. Leveque. - Topics in number theory, Vol II, Addison-Wesley, 1958.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • D. R. Heath-Brown
    • 1
  1. 1.Magdalen CollegeOxfordGrande Bretagne

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