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

Abstract:

We propose a new search algorithm to solve the equation $x^3+y^3+z^3=n$ for a fixed value of $n>0$. By parametrizing $|x|=$min$(|x|, |y|, |z|)$, this algorithm obtains $|y|$ and $|z|$ (if they exist) by solving a quadratic equation derived from divisors of $|x|^3 \pm n$. By using several efficient number-theoretic sieves, the new algorithm is much faster on average than previous straightforward algorithms. We performed a computer search for 51 values of $n$ below 1000 (except $n\equiv \pm 4 (\operatorname {mod}9)$) for which no solution has previously been found. We found eight new integer solutions for $n=75, 435, 444, 501, 600, 618, 912,$ and $969$ in the range of $|x| \le 2 \cdot 10^7$.