A356711 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

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

Offset: 1

Author

Jianing Song, Aug 23 2022

Keywords

• nonn
• hard
• more

Comments

Cube root of A179149.

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).

Examples

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

Crossrefs

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356712.

Indices of 5 in A356706, of 2 in A356707, and of 3 in A356708.

Extensions

a(31)-a(35) from Max Alekseyev, Jun 01 2023