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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].

MSC:

11Y50 Computer solution of Diophantine equations
11P05 Waring’s problem and variants
11D25 Cubic and quartic Diophantine equations
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