A336230 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 128.

1, 4, 9, 25, 49, 81, 121, 169, -224, 225, 289, 361, -383, 441, 504, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2025, 2209, -2219, 2401, 2601, -2687, 2809, 3025, 3249, 3481, -3680, 3721, 3969, 4225, -4283, 4417, 4489, 4761, 5041, 5329, -5459

Offset

1,2

Comments

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

(5 - 4*n^2)^3 + (5 - 4*(n + 1)^2)^3 + 2*(2*n + 1)^6 = 128. A000290(2*n + 1) are terms of the sequence, i.e., there is an infinity of nontrivial solutions to the equation.

References

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

Links

Table of n, a(n) for n=1..47.

Example

1^3 + 5^3 + 2 * 1^3 = 128, 1 is a term.
(-11)^3 + (-31)^3 + 2 * (25)^3 = 128, 25 is a term.

Mathematica

Clear[t]
t = {};
Do[y = (128 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -4761, 4761}, {x, -11550, 11550}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 9523}];
Select[v, MemberQ[u, #] &]

Crossrefs

Cf. A000290, A000578, A003215, A004825, A004826, A050791, A130472, A195006.

Sequence in context: A332646 A306043 A194269 * A130283 A065739 A053704

Adjacent sequences: A336227 A336228 A336229 * A336231 A336232 A336233

Keyword

sign

Author

XU Pingya, Jul 12 2020

Status

approved