A372543 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x + 1 is n, or -1 if no such k exists.

0, 1, 2, 4, 8, 17, 61, 347, 3778, 11416

Offset

0,3

Comments

This family of curves quickly reaches a moderate value of rank with a relatively small parameter k.

By heuristic search (see links), a(10) <= 216493 and a(11) <= 1448203.

Links

Table of n, a(n) for n=0..9.

Anna Antoniewicz, On a family of elliptic curves, (2005) Iagellonicae Acta Mathematica, XLIII.

Jose Aranda, Non sequential search for upper bounds (PARI-GP Script)

Cecylia Bocovich, Elliptic Curves of High Rank, (2012) Macalester College, Science Honors Projects.

Prog

(PARI) a(n, startAt=0)=for(k=startAt, oo, my(t=ellrank(ellinit([-k^2, +1]))); if(t[2]<n, next); if(t[1]>n, warning("k=", k, " has rank in ", t[1..2]); next); if(t[1]<n || t[2]>n, error("Cannot determine if a(", n, ") is ", k, " or larger; rank is in ", t[1..2])); return(k)) \\ Charles R Greathouse IV, Jul 08 2024
(PARI) \\ See Aranda link.

Crossrefs

Cf. A194687, A309060, A309068, A309069, A309146, A309147, A309164, A309165.

Sequence in context: A231221 A231435 A113153 * A255999 A171719 A185178

Adjacent sequences: A372540 A372541 A372542 * A372544 A372545 A372546

Keyword

nonn,hard,more

Author

Jose Aranda, Jul 04 2024

Status

approved