A383047 Squarefree d such that x^3+y^3=z^3 has non-trivial solution in Q(sqrt(d)).

2, 5, 6, 11, 14, 15, 17, 23, 26, 29, 33, 35, 38, 41, 42, 43, 47, 51, 53, 58, 59, 62, 65, 69, 71, 74, 77, 78, 82, 83, 85, 86, 87, 89, 93, 95, 101, 105, 106, 107, 109, 110, 113, 114, 119, 122, 123, 131, 134, 137, 141, 142, 143, 146, 149, 155, 158, 159, 161, 167, 170

Offset

1,1

Comments

Equivalent condition is that the elliptic curve dY^2=X^3-432 has positive rank.

Under the Birch and Swinnerton-Dyer conjecture, d not divisible by 3 appears in this sequence if and only if x^2 + y^2 + 7z^2 + xz = d and x^2 + 2y^2 + 4z^2 + xy + yz = d have equal numbers of integral solutions, and d divisible by 3 appears in this sequence if and only if x^2 + 3y^2 + 27z^2 = d/3 and 3x^2 + 4y^2 + 7z^2 - 2yz = d/3 have equal numbers of integral solutions.

For d not divisible by 3, d appears in this sequence if and only if 3d appears in A383048, and 3d appears in this sequence if and only if d appears in A383048.

References

M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), no. 6, 1579-1591.

Links

Seiichi Azuma, Table of n, a(n) for n = 1..160

M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields.

Example

For a(1)=2, (18+17*sqrt(2))^3+(18-17*sqrt(2))^3=42^3.

Prog

(PARI) for(n=2, 500, if(vecmax(factor(n)[, 2])>= 2, next); r=ellrank(ellinit([0, 0, 0, 0, -432*n^3])); if(r[2]>0, print1(n, ", "); if(r[1]==0, print("uncertain!"))))

Crossrefs

Cf. A383048.

Sequence in context: A092306 A349158 A383048 * A319242 A323398 A233865

Adjacent sequences: A383044 A383045 A383046 * A383048 A383049 A383050

Keyword

nonn

Author

Seiichi Azuma, Apr 14 2025

Status

approved