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%I #89 Aug 16 2024 21:03:22
%S 0,1,2,4,8,17,61,347,3778,11416
%N 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.
%C This family of curves quickly reaches a moderate value of rank with a relatively small parameter k.
%C By heuristic search (see links), a(10) <= 216493 and a(11) <= 1448203.
%H Anna Antoniewicz, <a href="http://www.emis.de/journals/UIAM/PDF/43-21-32.pdf">On a family of elliptic curves</a>, (2005) Iagellonicae Acta Mathematica, XLIII.
%H Jose Aranda, <a href="/A372543/a372543.txt">Non sequential search for upper bounds</a> (PARI-GP Script)
%H Cecylia Bocovich, <a href="https://digitalcommons.macalester.edu/mathcs_honors/24">Elliptic Curves of High Rank</a>, (2012) Macalester College, Science Honors Projects.
%o (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
%o (PARI) \\ See Aranda link.
%Y Cf. A194687, A309060, A309068, A309069, A309146, A309147, A309164, A309165.
%K nonn,hard,more
%O 0,3
%A _Jose Aranda_, Jul 04 2024