A336166 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 2.

0, 1, -3, 4, 9, -12, 16, 25, -27, -35, 36, 37, -48, 49, -59, 64, -75, 81, 100, -108, 121, 144, -147, -159, 169, 172, -192, 196, 225, -227, -243, -255, 256, 261, -287, 289, -300, -311, 324, -335, 361, -363, 373, 400, -432, 441, 484, -507, 529, 568, 576, -588

Offset: 1

Author

XU Pingya, Jul 10 2020

Keywords

• sign

Comments

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

Segre shows that 1-(9/2)*A000578(2n), (-3)*A000290(n), and A016754(n) are terms of the sequence.

Examples

			(-5)^3 + (-11)^3 + 2 * 9^3 = 2, 9 is a term.
(25)^3 + (-23)^3 + 2 * (-12)^3 = 2, -12 is a term.
		

References

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

Links

• Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.

Crossrefs

Cf. A000290, A000578, A004825, A004826, A016754, A028387, A050791, A130472, A195006.

Programs

Mathematica

t1 = Union[Plus@@@Tuples[Range[-11643, 11643]^3, 2]];
t2 = Table[2 - 2z^3, {z, -588, 588}];
t = Select[t1, MemberQ[t2, #] &];
u = ((2 - t)/2)^(1/3) /. (-1)^(1/3) -> (-1);
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 1176}];
Select[v, MemberQ[u, #] &]