On searching for solutions of the Diophantine equation

Author:
Kenji Koyama

Journal:
Math. Comp. **69** (2000), 1735-1742

MSC (1991):
Primary 11D25

DOI:
https://doi.org/10.1090/S0025-5718-00-01202-3

Published electronically:
February 21, 2000

MathSciNet review:
1680899

Full-text PDF

Abstract | References | Similar Articles | Additional Information

We propose an efficient search algorithm to solve the equation for a fixed value of . By parametrizing , this algorithm obtains and (if they exist) by solving a quadratic equation derived from divisors of . Thanks to the use of several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for six values of below 1000 for which no solution had previously been found. We found three new integer solutions for and 931 in the range of .

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Additional Information

**Kenji Koyama**

Affiliation:
NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02 Japan

Email:
koyama@cslab.kecl.ntt.co.jp

DOI:
https://doi.org/10.1090/S0025-5718-00-01202-3

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

Received by editor(s):
October 7, 1996

Received by editor(s) in revised form:
January 18, 1999

Published electronically:
February 21, 2000

Article copyright:
© Copyright 2000
American Mathematical Society