A179149 Numbers k such that Mordell's equation y^2 = x^3 + k has exactly 5 integral solutions.

1, 64, 729, 1000, 2744, 4096, 15625, 21952, 35937, 46656, 50653, 64000, 117649, 262144, 343000, 531441, 592704, 681472, 729000, 753571, 1000000, 1124864, 1771561, 2000376, 2197000, 2299968, 2744000, 2985984, 3652264, 4096000, 4826809, 5451776, 6229504, 7189057, 7529536

Offset

1,2

Comments

Contains all sixth powers: 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). - Jianing Song, Aug 24 2022

Links

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

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Formula

a(n) = A356711(n)^3.

Crossrefs

Cf. A054504, A081119, A179145-A179162, A356711.

Sequence in context: A048392 A274219 A161861 * A074471 A164345 A164337

Adjacent sequences: A179146 A179147 A179148 * A179150 A179151 A179152

Keyword

nonn

Author

Artur Jasinski, Jun 30 2010

Extensions

Edited and extended by Ray Chandler, Jul 11 2010

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

Status

approved