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

%I A383047 #13 Apr 21 2025 23:08:13
%S A383047 2,5,6,11,14,15,17,23,26,29,33,35,38,41,42,43,47,51,53,58,59,62,65,69,
%T A383047 71,74,77,78,82,83,85,86,87,89,93,95,101,105,106,107,109,110,113,114,
%U A383047 119,122,123,131,134,137,141,142,143,146,149,155,158,159,161,167,170
%N A383047 Squarefree d such that x^3+y^3=z^3 has non-trivial solution in Q(sqrt(d)).
%C A383047 Equivalent condition is that the elliptic curve dY^2=X^3-432 has positive rank.
%C A383047 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.
%C A383047 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.
%D A383047 M. Jones and J. Rouse, Solutions of the cubic Fermat equation in quadratic fields, Int. J. Number Theory 9 (2013), no. 6, 1579-1591.
%H A383047 Seiichi Azuma, <a href="/A383047/b383047.txt">Table of n, a(n) for n = 1..160</a>
%H A383047 M. Jones and J. Rouse, <a href="https://users.wfu.edu/rouseja/cv/jonesrouse.pdf">Solutions of the cubic Fermat equation in quadratic fields</a>.
%e A383047 For a(1)=2, (18+17*sqrt(2))^3+(18-17*sqrt(2))^3=42^3.
%o A383047 (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!"))))
%Y A383047 Cf. A383048.
%K A383047 nonn
%O A383047 1,1
%A A383047 _Seiichi Azuma_, Apr 14 2025