On searching for solutions of the Diophantine equation
Authors:
Kenji Koyama, Yukio Tsuruoka and Hiroshi Sekigawa
Journal:
Math. Comp. 66 (1997), 841851
MSC (1991):
Primary 11D25
MathSciNet review:
1401942
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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 numbertheoretic 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\ 22 Hikaridai, Seikacho, Sorakugun, Kyoto 61902 Japan
Email:
koyama@cslab.kecl.ntt.jp
Yukio Tsuruoka
Affiliation:
NTT Communication Science Laboratories\ 22 Hikaridai, Seikacho, Sorakugun, Kyoto 61902 Japan
Email:
tsuruoka@cslab.kecl.ntt.jp
Hiroshi Sekigawa
Affiliation:
NTT Communication Science Laboratories\ 22 Hikaridai, Seikacho, Sorakugun, Kyoto 61902 Japan
Email:
sekigawa@cslab.kecl.ntt.jp
DOI:
http://dx.doi.org/10.1090/S0025571897008302
PII:
S 00255718(97)008302
Keywords:
Diophantine equation,
cubic,
numbertheoretic 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
