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a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.
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%I #21 Jul 06 2024 11:10:06

%S 1,2,11,113,2089,28279,975379

%N a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

%C a(n) = min{ A031507(n), A031508(n) }.

%C See A031507 and A031508 for further information.

%C a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - _N. J. A. Sloane_, Jul 05 2024

%D Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

%H Noam D. Elkies and Zev Klagsbrun, <a href="https://doi.org/10.2140/obs.2020.4.233">New rank records for elliptic curves having rational torsion</a>, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.

%Y Cf. A031507, A031508, A081119.

%K nonn,more,hard

%O 0,2

%A _N. J. A. Sloane_, Jul 04 2024