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On searching for solutions of the Diophantine equation
Authors:Kenji Koyama  Yukio Tsuruoka  Hiroshi Sekigawa
Institution:NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan ; NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan ; NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Abstract:We propose a new search algorithm to solve the equation $x^3+y^3+z^3=n$ for a fixed value of $n>0$. By parametrizing $|x|=$min$(|x|, |y|, |z|)$, this algorithm obtains $|y|$ and $|z|$ (if they exist) by solving a quadratic equation derived from divisors of $|x|^3 \pm n$. By using several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for 51 values of $n$ below 1000 (except $n\equiv \pm 4 (\MOD 9)$) for which no solution has previously been found. We found eight new integer solutions for $n=75, 435, 444, 501, 600, 618, 912,$ and $969$ in the range of $|x| \le 2 \cdot 10^7$.

Keywords:Diophantine equation  cubic  number-theoretic sieves  search algorithm  computer search
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