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%I #34 Jul 01 2024 12:41:56
%S 1,5,34,1254,29274,48272239,6611719866
%N Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x is n, or -1 if no such k exists.
%C Fermat found a(0), Biling found a(1), and Wiman found a(2)-a(4). Rogers found upper bounds on a(5) and a(6) equal to their true value; Rathbun and an unknown author verified them as a(5) and a(6), respectively.
%C a(7) <= 797507543735, see Rogers 2004.
%D G. Billing, "Beiträge zur arithmetischen theorie der ebenen kubischen kurven geschlechteeins", Nova Acta Reg. Soc. Sc. Upsaliensis (4) 11 (1938), Nr. 1. Diss. 165 S.
%D N. Rogers, "Elliptic curves x^3 + y^2 = k with high rank", PhD Thesis in Mathematics, Harvard University (2004).
%D A. Wiman, "Über rationale Punkte auf Kurven y^2 = x(x^2-c^2)", Acta Math. 77 (1945), pp. 281-320.
%H Andrej Dujella, Ali S. Janfada, and Sajad Salami, <a href="http://emis.mi.ras.ru/journals/JIS/VOL12/Janfada/janfada3.html">A search for high rank congruent number elliptic curves</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.8.
%H Randall L. Rathbun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;16b991d8.1108">Posting to NMBRTHRY</a>, Aug 25 2011
%H N. F. Rogers, <a href="http://www.emis.de/journals/EM/restricted/9/9.4/rogers.ps">Rank computations for the congruent number elliptic curves</a>, Exper. Math. 9:4 (2000), pp. 591-594.
%H K. Rubin and A. Silverberg, <a href="http://www.ams.org/bull/2002-39-04/S0273-0979-02-00952-7/home.html">Ranks of elliptic curves</a>, p.464, Table 2.
%H Mark Watkins, <a href="http://magma.maths.usyd.edu.au/~watkins/papers/r3.pdf">On elliptic curves and random matrix theory</a>, Journal de Theorie des Nombres de Bordeaux
%H Author?, <a href="http://wiki.l-functions.org/LfunctionsAndModularFormsII/CentralValues/Rank4">LfunctionsAndModularFormsII / CentralValues / Rank4</a>
%o (PARI) r(n)=ellanalyticrank(ellinit([0,0,0,-n^2,0]))[1]
%o rec=0;for(n=1,1e4,t=r(n);if(t>rec,rec=t;print("r("n") = "t)))
%Y Cf. A062693, A062695, A003273, A309028, A309029, A319510.
%K nonn,hard,more
%O 0,2
%A _Charles R Greathouse IV_, Sep 01 2011
%E Escape clause added to definition by _N. J. A. Sloane_, Jul 01 2024