A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808

Offset: 1

Author

XU Pingya, Nov 16 2020

Keywords

• nonn

Comments

The data are derived from the following formula:

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

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

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

(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;

(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;

(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;

(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.

Examples

			16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.
		

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.

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.

Programs

Mathematica

t1 = 2*Range[23]^6;
t2 = 2*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*Range[4]^9;
t4 = 2*{1, 5}^12;
t5 = 16*{1, 2, 4}^12;
t6 = 1458*{1, 3}^12;
t7 = 11664*{1, 2, 3}^12;
Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]

Extensions

Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022