Remote Access Mathematics of Computation
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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 $\vert z\vert \le 5 \cdot 10^7$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11D25

Retrieve articles in all journals with MSC (1991): 11D25

Additional Information

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

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