A383047 -id:A383047 - cp's OEIS frontend

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

Seiichi Azuma, Apr 14 2025

nonn

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.

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

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

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.

Cf. A383047.

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!"))))

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

cp's OEIS Frontend

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

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