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A179149
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Numbers k such that Mordell's equation y^2 = x^3 + k has exactly 5 integral solutions.
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8
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1, 64, 729, 1000, 2744, 4096, 15625, 21952, 35937, 46656, 50653, 64000, 117649, 262144, 343000, 531441, 592704, 681472, 729000, 753571, 1000000, 1124864, 1771561, 2000376, 2197000, 2299968, 2744000, 2985984, 3652264, 4096000, 4826809, 5451776, 6229504, 7189057, 7529536
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OFFSET
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1,2
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COMMENTS
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Contains all sixth powers: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3). - Jianing Song, Aug 24 2022
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LINKS
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FORMULA
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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