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A031508
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Smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank n, if k exists.
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6
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OFFSET
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0,2
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COMMENTS
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The sequence might be finite, even if it is redefined as smallest k>0 such that the elliptic curve y^2 = x^3 - k has rank >= n. - Jonathan Sondow, Sep 26 2013
For bounds on later terms see the Gebel link. - N. J. A. Sloane, Jul 05 2010
See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n. - Jonathan Sondow, Sep 06 2013
Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - Charles R Greathouse IV, Sep 10 2013
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LINKS
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PROG
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(PARI) {a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019
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CROSSREFS
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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