On searching for solutions of

the Diophantine equation

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 | References | Similar Articles | Additional Information

Abstract: We propose a new search algorithm to solve the equation for a fixed value of . By parametrizing min, this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . 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 below 1000 (except ) for which no solution has previously been found. We found eight new integer solutions for and in the range of .

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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