Searching for Solutions of x3 + y3 + z3 = k
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  in 1964 failed to resolve these questions.
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