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A194687
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Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x is n.
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5
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OFFSET
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0,2
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COMMENTS
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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.
a(7) <= 797507543735, see Rogers 2004.
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REFERENCES
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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.
N. Rogers, "Elliptic curves x^3 + y^2 = k with high rank", PhD Thesis in Mathematics, Harvard University (2004).
A. Wiman, "Über rationale Punkte auf Kurven y^2 = x(x^2-c^2)", Acta Math. 77 (1945), pp. 281-320.
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LINKS
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PROG
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(PARI) r(n)=ellanalyticrank(ellinit([0, 0, 0, -n^2, 0]))[1]
rec=0; for(n=1, 1e4, t=r(n); if(t>rec, rec=t; print("r("n") = "t)))
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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STATUS
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approved
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