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Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.
4

%I #59 Jul 02 2024 02:10:56

%S 1,3,17,627,14637

%N Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.

%C From _Jose Aranda_, Jun 30 2024: (Start)

%C A319510(n even) = A309061(n/2), A319510(n odd) = A309061(2*n) (Empirical).

%C A194687(5) = 48272239 which implies a(5) <= 96544478 (Checked).

%C A194687(6) = 6611719866 which implies a(6) <= 3305859933 (Checked).

%C A194687(7) <= 797507543735 which implies a(7) <= 1595015087470 (Checked). (End)

%F A309061(a(n)) = n.

%e A309061(1) = 0.

%e A309061(3) = 1.

%e A309061(17) = 2.

%o (PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, k^2, 0]))[1]<>n, k++); k}

%Y Cf. A194687, A309028, A309061, A319510.

%K nonn,more,hard

%O 0,2

%A _Seiichi Manyama_, Jul 09 2019