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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 $x\sp 3+ y\sp 3+ z\sp 3=k$, for small values of $k$. Representations for $k= 39$, 84, 556 and 870 were discovered. Previous works [see {\it V. L. Gardiner}, {\it R. B. Lazarus} and {\it P. R. Stein}, Math. Comput. 18, 408-413 (1964; Zbl 0121.284), for example] had failed to find solutions for these $k$. A solution for $k=2$, not belonging to the family $6t\sp 3+1$, $-6t\sp 3+1$, $-6t\sp 2$ has also been found. Some of these results had been found independently by the reviewer, {\it W. M. Lioen} and {\it 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 $z\sp 3\equiv k \pmod {x+y}$ 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 $\vert x\vert,\vert y\vert,\vert z\vert\leq 2\cdot 2\times 10\sp 6$. This compares with the cube $\vert x\vert,\vert y\vert,\vert z\vert\leq 1\cdot 3\times 10\sp 8$ 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