Abstract
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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References
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.
W.J. Leveque. - Topics in number theory, Vol II, Addison-Wesley, 1958.
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Heath-Brown, D.R. (1992). Searching for Solutions of x 3 + y 3 + z 3 = k . In: David, S. (eds) Séminaire de Théorie des Nombres, Paris, 1989–90. Progress in Mathematics, vol 102. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-4269-5_6
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DOI: https://doi.org/10.1007/978-1-4757-4269-5_6
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