A179175 a(n) = least positive k such that Mordell's equation y^2 = x^3 - k has exactly n integral solutions.

3, 1, 2, 1331, 4, 216, 28, 54872, 116, 343, 828, 250047, 496, 71991296, 207

Offset: 0

Author

Artur Jasinski, Jun 30 2010

Keywords

• nonn
• hard
• more

Comments

The status of further terms is:

15 integral solutions: unknown

16 integral solutions: 503

17 integral solutions: unknown

18 integral solutions: 431

19 integral solutions: unknown

20 integral solutions: 2351

21 integral solutions: unknown

22 integral solutions: 3807

For least positive k such that equation y^2 = x^3 + k has exactly n integral solutions, see A179162.

If n is odd, then a(n) is perfect cube. [Ray Chandler]

From Jose Aranda, Aug 04 2024: (Start)

About those unknown terms:

a(15) <= 2600^3 = (26* 10^2)^3

a(17) <= 10400^3 = (26* 20^2)^3

a(19) <= 93600^3 = (26* 60^2)^3

a(21) <= 4586400^3 = (26*420^2)^3

The term a(13) = 71991296 = 416^3 = (26*4^2)^3. (End)

Links

• 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]

Crossrefs

Cf. A081120, A081121, A179163-A179174.

Extensions

Edited and a(7), a(11), a(13) added by Ray Chandler, Jul 11 2010