A383048 -id:A383048 - cp's OEIS frontend

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

Seiichi Azuma, Apr 14 2025

nonn

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.

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

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..160
M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields.

Cf. A383048.

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!"))))

cp's OEIS Frontend

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

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