A338932 Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.
1, 2, 128, 729, 1458, 4096, 65536, 93312, 2985984, 3906250, 16777216, 28697814, 33554432, 47775744, 80707214, 244140625, 250000000, 387420489, 1836660096, 2847656250, 4715895382, 5165261696, 12230590464, 13841287201, 17179869184, 21208998746, 24461180928
Offset: 1
Keywords
Examples
128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.
References
- R. K. Guy, Unsolved Problems in Number Theory, D5.
Links
- Kenji Koyama, On searching for solutions of the Diophantine equation x^3 + y^3 + 2z^3 = n, Math. Comp, 69 (2000), 1735-1742.
- J. C. P. Miller & M. F. C. Woollett, Solutions of the Diophantine equation x^3 + y^3 + z^3 = k, J. London Math. Soc. 30(1955), 101-110.
- Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.
Programs
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Mathematica
t1 = 2*{1, 5, 7, 11, 13}^9; t2 = 128*{1, 2, 4, 5, 7, 8}^9; t3 = 1458*{1, 3, 5, 7, 9}^9; t4 = 93312*{1, 2, 3, 4, 5}^9; t5 = {1, 2, 4, 5, 7}^12; t6 = 729*{1, 2, 3, 4, 5}^12; Take[Union[t1, t2, t3, t4, t5, t6], 27]
Comments