On searching for solutions of the Diophantine equation 𝑥³+𝑦³+2𝑧³=𝑛

Koyama, Kenji

doi:10.1090/S0025-5718-00-01202-3

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

Math. Comp. 69 (2000), 1735-1742

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

Published electronically: February 21, 2000

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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 $|z|$, this algorithm obtains $|x|$ and $|y|$ (if they exist) by solving a quadratic equation derived from divisors of $2|z|^3 \pm n$. 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 $|z| \le 5 \cdot 10^7$.

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