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A031507
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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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11
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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
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
The curves for k and -27*k are isogenous (as Noam Elkies points out---see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
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LINKS
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FORMULA
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EXAMPLE
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a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack). - Jonathan Sondow, Sep 10 2013
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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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