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A356707
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Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y positive.
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7
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2, 3, 0, 2, 0, 0, 1, 4, 2, 2, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 1, 1, 1, 0, 2, 1, 0, 2, 0, 0, 0, 4, 2, 1, 1, 2, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 2, 3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 1, 2, 8, 0, 0, 0, 0, 2, 1, 4, 0, 1, 0, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 0, 1, 0, 0, 3, 1, 2
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OFFSET
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1,1
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COMMENTS
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Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3 apart from the trivial solution (-n,0).
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LINKS
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FORMULA
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EXAMPLE
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a(2) = 3 because the solutions to y^2 = x^3 + 2^3 with y > 0 are (1,3), (2,4), and (46,312).
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PROG
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(SageMath) [(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))-1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022
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CROSSREFS
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KEYWORD
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nonn,hard
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AUTHOR
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EXTENSIONS
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Offset and a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 03 2022
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STATUS
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approved
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