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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On searching for solutions of the Diophantine equation $x^3 + y^3 +2z^3 = n$


Author: Kenji Koyama
Journal: Math. Comp. 69 (2000), 1735-1742
MSC (1991): Primary 11D25
Published electronically: February 21, 2000
MathSciNet review: 1680899
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Abstract:

We propose an efficient search algorithm to solve the equation $x^3+y^3+ 2z^3=n$ for a fixed value of $n>0$. By parametrizing $\vert z\vert$, this algorithm obtains $\vert x\vert$ and $\vert y\vert$ (if they exist) by solving a quadratic equation derived from divisors of $2\vert z\vert^3 \pm n$. 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 $n$ below 1000 for which no solution had previously been found. We found three new integer solutions for $n=183, 491$ and 931 in the range of $\vert z\vert \le 5 \cdot 10^7$.


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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: http://dx.doi.org/10.1090/S0025-5718-00-01202-3
PII: S 0025-5718(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