

A356711


Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.


8



1, 4, 9, 10, 14, 16, 25, 28, 33, 36, 37, 40, 49, 64, 70, 81, 84, 88, 90, 91, 100, 104, 121, 126, 130, 132, 140, 144, 154, 160, 169, 176, 184, 193, 196
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OFFSET

1,2


COMMENTS

Contains all squares: 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).


LINKS



EXAMPLE

1 is a term since the equation y^2 = x^3 + 1^3 has 5 solutions (1,0), (0,+1), and (2,+3).


CROSSREFS



KEYWORD

nonn,hard,more


AUTHOR



EXTENSIONS



STATUS

approved



