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

1,2

Comments

The data are derived from the following formula:

(a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;

(4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;

(9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;

(36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;

((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;

((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.

References

R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Table of n, a(n) for n=1..27.

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.

Example

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.

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]

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338933.

Sequence in context: A157473 A004864 A106319 * A226403 A106320 A106318

Adjacent sequences: A338929 A338930 A338931 * A338933 A338934 A338935

Keyword

nonn

Author

XU Pingya, Nov 16 2020

Status

approved