@MISC{Garaev_third-degreediophantine, author = {M.Z. Garaev}, title = {Third-Degree Diophantine Equations}, year = {} }

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Abstract

INTRODUCTION This work is devoted to the investigation of the solvability of the Diophantine equation x 3 + y 3 + z 3 = nxyz; (1) where n = 2 f0; 3g, n is an integer. The rst results were obtained as long ago as 1856 by Sylvester (see [1]). He proved that for n = 6 (1) had no solutions in nonzero integers. The next results were obtained almost eighty years later. In 1933-34 Hurwitz and Mordell showed that for n = 1 and for n = 5 (1) had the only solutions (1; 1; 1) and (1; 1; 2), respectively, with an accuracy to within the permutation and to the common factor (see [2, 3]). Developing Mordell's method, Cassels showed in 1960 that for n = 1 (1) had no solutions in