A356711 - OEIS

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

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OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..35.

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

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

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

Sequence in context: A208980 A010426 A243170 * A054294 A198097 A358762

Adjacent sequences: A356708 A356709 A356710 * A356712 A356713 A356714

KEYWORD

nonn,hard,more

AUTHOR

Jianing Song, Aug 23 2022

EXTENSIONS

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

STATUS

approved