A383048 Squarefree d such that x^3+y^3=z^3 has nontrivial solution in Q(sqrt(-d)).

2, 5, 6, 11, 14, 15, 17, 23, 26, 29, 31, 33, 35, 38, 41, 42, 47, 51, 53, 59, 62, 65, 69, 71, 74, 77, 78, 83, 86, 87, 89, 95, 101, 105, 106, 107, 109, 110, 113, 114, 119, 122, 123, 129, 131, 134, 137, 141, 143, 146, 149, 155, 158, 159, 161, 167, 170, 173, 174, 177, 179

Offset

1,1

Comments

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

If d == 1 (mod 3) appears in this sequence, 3 divides the class number of Q(sqrt(-d)).

For d not divisible by 3, d appears in this sequence if and only if 3d appears in A383047, and 3d appears in this sequence if and only if d appears in A383047.

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..168

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

Example

For a(1)=2, (-2+sqrt(-2))^3+(-2-sqrt(-2))^3=2^3

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. A383047.

Sequence in context: A002133 A092306 A349158 * A383047 A319242 A323398

Adjacent sequences: A383045 A383046 A383047 * A383049 A383050 A383051

Keyword

nonn

Author

Seiichi Azuma, Apr 14 2025

Extensions

a(50) corrected by David Radcliffe, Aug 01 2025

Status

approved