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We propose a new search algorithm to solve the equation x3+y3 + z3 = 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 ± 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 ≡ ±4 (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| ≤ 2 · 107.

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