Searching for Solutions of x ³ + y ³ + z ³ = k

Abstract

Diophantine equations of the form

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa % aaleqabaGaaG4maaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaioda % aaGccqGHRaWkcaWG6bWaaWbaaSqabeaacaaIZaaaaOGaeyypa0Jaam % 4Aaaaa!3F85!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${x^3} + {y^3} + {z^3} = k$$

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.