Conn, W.; Vaserstein, L. N. On sums of three integral cubes. (English) Zbl 0808.11075 Andrews, George E. (ed.) et al., The Rademacher legacy to mathematics. The centenary conference in honor of Hans Rademacher, July 21-25, 1992, Pennsylvania State University, University Park, PA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 166, 285-294 (1994). The authors report on calculations to find integral solutions of \(x^ 3+ y^ 3+ z^ 3=k\), for small values of \(k\). Representations for \(k= 39\), 84, 556 and 870 were discovered. Previous works [see V. L. Gardiner, R. B. Lazarus and P. R. Stein, Math. Comput. 18, 408–413 (1964; Zbl 0121.28402), for example] had failed to find solutions for these \(k\). A solution for \(k=2\), not belonging to the family \(6t^ 3+1\), \(-6t^ 3+1\), \(-6t^ 2\) has also been found. Some of these results had been found independently by the reviewer, W. M. Lioen and H. J. J. te Riele [Math. Comput. 61, 235–244 (1993; Zbl 0783.11046)]. Different search strategies were used for different ranges of the variables, but in each case \(x+y\) was first fixed, and the congruence \(z^ 3\equiv k \pmod {x+y}\) solved. The search region was somewhat complicated, and the largest solutions found had 10 digits. However the largest cube in the search region is roughly \(| x|,| y|,| z|\leq 2\cdot 2\times 10^ 6\). This compares with the cube \(| x|,| y|,| z|\leq 1\cdot 3\times 10^ 8\) covered by the reviewer et al. (loc. cit.) for the case \(k=3\). Unfortunately the present paper does not specify the amount of computer time used.The paper concludes with the conjecture that, for fixed \(k\not\equiv \pm 4\pmod 9\), the number of solutions in a cube of side \(M\) grows as \(c(k)\log M\). This agrees with the heuristics presented by the reviewer [Math. Comput. 59, 613–623 (1992; Zbl 0778.11017)].For the entire collection see [Zbl 0798.00010]. Reviewer: D. R. Heath-Brown (Oxford) Cited in 3 Documents MSC: 11Y50 Computer solution of Diophantine equations 11P05 Waring’s problem and variants 11D25 Cubic and quartic Diophantine equations Keywords:computer search; representation of integers; sums of cubes; cubic diophantine equation Citations:Zbl 0121.28402; Zbl 0783.11046; Zbl 0778.11017 PDFBibTeX XMLCite \textit{W. Conn} and \textit{L. N. Vaserstein}, Contemp. Math. 166, 285--294 (1994; Zbl 0808.11075) Online Encyclopedia of Integer Sequences: Value of x of the solution to x^3 + y^3 + z^3 = A060464(n) (numbers not 4 or 5 mod 9) with smallest |z| and smallest |y|, 0 <= |x| <= |y| <= |z|.