On solving the Diophantine equation $x^ 3+y^ 3+z^ 3=k$ on a vector computer

Abstract:

Recently, the first author has proposed a new algorithm for solving the Diophantine equation ${x^3} + {y^3} + {z^3} = k$, where k is a given nonzero integer. In this paper we present the detailed versions of this algorithm for some values of k given below, and we describe how we have optimized and run the algorithm on a Cyber 205 vector computer. A vectorized version of the Euclidean algorithm was written, which is capable of solving the equations ${w_i}{x_i} \equiv 1 \bmod {n_i},i = 1,2, \ldots$, at vector speed. Moreover, the basic double-precision arithmetic operations $( + , - , \times ,/)$ were vectorized. The following cases were implemented and run: $k = 3,30,2,20,39$, and 42. For $k = 3$ a range was searched which includes the cube $|x|,|y|,|z| \leq {10^8}$; this considerably extends an earlier search in the cube $|x|,|y|,|z| \leq {2^{16}}$. No solutions were found apart from the known ones (1, 1, 1) and $(4,4, - 5)$. For $k = 30$, which probably is, with $k = 3$, the case which has attracted most attention in the literature, no solution was found. It is the smallest case for which no solution is known and for which one has not been able to find a proof that no solution exists. For $k = 2$ a parametric form solution is known, but we also found one which does not belong to this parametric form, viz., $(1214928,3480205, - 3528875)$. For $k = 20$, several new large solutions were found in addition to several known ones; this case served as a (partial) check of the correctness of our program. Finally, for $k = 39$ we found the first solution: $(134476,117367, - 159380)$. Hence, this case can be dropped from the list of values of k $(k = 30,33,39,42, \ldots )$ for which no solution is known (yet).