A356713 Numbers k such that Mordell's equation y^2 = x^3 - k^3 has exactly 1 integral solution.

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 1

Author

Jianing Song, Aug 23 2022

Keywords

• nonn

Comments

Numbers k such that Mordell's equation y^2 = x^3 - k^3 has no solution other than the trivial solution (k,0).

Cube root of A179163.

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), so y^2 = x^3 - t^6 has only one solution (t^2,0).

Links

• Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)

Crossrefs

Cf. A081120, A179163, A356709, A356720. Complement of A228948.

Formula

1 is a term since the equation y^2 = x^3 - 1^3 has no solution other than (1,0).