On searching for solutions of the Diophantine equation \(x^3 + y^3 +2z^3 = n\). (English) Zbl 0964.11058
It is conjectured that every positive integer \(n\) can be represented as \(n=x^3+ y^3+2z^3\), with \(x,y,z\in\mathbb{Z}\). Previous researchers have found solutions for all \(n\leq 1000\) except for 148, 183, 491, 671, 788 and 931. The present paper describes a search for solutions with \(|z|\leq 5 \times 10^7\). Solutions were found for \(n=183\), 491 and 931. For the remaining \(n\) there are no solutions with \(|z|\leq 5\times 10^7\). The computation took about one month of CPU time. The algorithm employed requires one to factor \(n-2z^3\) for each suitable \(z\) in the search region. The process is said to take time \(O(L^2)\) to search the range \(|z|\leq L\), with a small implied constant. This is slower in theory than the method of Elkies, which has yet to be applied to this problem.
Reviewer: Roger Heath-Brown (Oxford)
Keywords:
easier Waring problemOnline Encyclopedia of Integer Sequences:
Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.
References:
| [1] | J. H. E. Cohn, private communication (1995). |
| [2] | V. A. Dem\(^{\prime}\)janenko, Sums of four cubes, Izv. Vysš. Učebn. Zaved. Matematika 1966 (1966), no. 5 (54), 64 – 69 (Russian). |
| [3] | Richard K. Guy, Unsolved problems in number theory, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. · Zbl 0474.10001 |
| [4] | Richard K. Guy, Unsolved problems in number theory, 2nd ed., Problem Books in Mathematics, Springer-Verlag, New York, 1994. Unsolved Problems in Intuitive Mathematics, I. · Zbl 0805.11001 |
| [5] | D. R. Heath-Brown, W. M. Lioen, and H. J. J. te Riele, On solving the Diophantine equation \?³+\?³+\?³=\? on a vector computer, Math. Comp. 61 (1993), no. 203, 235 – 244. · Zbl 0783.11046 |
| [6] | K. Koyama, Tables of solutions of the Diophantine equation \(x^3+y^3+ z^3=n\), Math. Comp. 62 (1994), 941-942. |
| [7] | Kenji Koyama, Yukio Tsuruoka, and Hiroshi Sekigawa, On searching for solutions of the Diophantine equation \?³+\?³+\?³=\?, Math. Comp. 66 (1997), no. 218, 841 – 851. · Zbl 0880.11028 |
| [8] | R. F. Lukes, private communication (1995). |
| [9] | L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. · Zbl 0188.34503 |
| [10] | Hiroshi Sekigawa and Kenji Koyama, Nonexistence conditions of a solution for the congruence \?^{\?}\(_{1}\)+\cdots+\?^{\?}_{\?}\equiv \?\pmod{\?\(^{n}\)}, Math. Comp. 68 (1999), no. 227, 1283 – 1297. · Zbl 0924.11025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
