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Mathematics of Computation

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On searching for solutions of
the Diophantine equation $x^3 + y^3 +z^3 = n$


Authors: Kenji Koyama, Yukio Tsuruoka and Hiroshi Sekigawa
Journal: Math. Comp. 66 (1997), 841-851
MSC (1991): Primary 11D25
DOI: https://doi.org/10.1090/S0025-5718-97-00830-2
MathSciNet review: 1401942
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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 $|x| \le 2 \cdot 10^7$.


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Additional Information

Kenji Koyama
Affiliation: NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Email: koyama@cslab.kecl.ntt.jp

Yukio Tsuruoka
Affiliation: NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Email: tsuruoka@cslab.kecl.ntt.jp

Hiroshi Sekigawa
Affiliation: NTT Communication Science Laboratories 2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan
Email: sekigawa@cslab.kecl.ntt.jp

DOI: https://doi.org/10.1090/S0025-5718-97-00830-2
Keywords: Diophantine equation, cubic, number-theoretic sieves, search algorithm, computer search
Received by editor(s): November 13, 1995
Received by editor(s) in revised form: February 14, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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