A356706 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3.

5, 7, 1, 5, 1, 1, 3, 9, 5, 5, 3, 1, 1, 5, 1, 5, 1, 7, 1, 1, 3, 3, 3, 1, 5, 3, 1, 5, 1, 1, 1, 9, 5, 3, 3, 5, 5, 3, 1, 5, 1, 1, 1, 3, 1, 3, 1, 1, 5, 7, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 1, 3, 5, 17, 1, 1, 1, 1, 5, 3, 9, 1, 3, 1, 1, 1, 9, 1, 1, 5, 1, 1, 5, 1, 3, 1, 5, 1, 5, 5, 3, 1, 1, 3, 1, 1

Offset: 1

Author

Jianing Song, Aug 23 2022

Keywords

• nonn
• hard

Examples

			a(8) = 9 since the equation y^2 = x^3 + 8^3 has 9 integral solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496).
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A081119, A356707, A356708.

Indices of 1, 3, 5, and 7: A356709, A356710, A356711, A356712.

Programs

SageMath

[len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True)) for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022

Formula

a(n) = A081119(n^3).

Extensions

a(21) corrected and a(22)-a(60) from Lucas A. Brown, Sep 03 2022

Terms a(61) onward from Max Alekseyev, Jun 01 2023