%I #24 Aug 04 2024 20:48:41

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

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

%C The status of further terms is:

%C 15 integral solutions: unknown

%C 16 integral solutions: 503

%C 17 integral solutions: unknown

%C 18 integral solutions: 431

%C 19 integral solutions: unknown

%C 20 integral solutions: 2351

%C 21 integral solutions: unknown

%C 22 integral solutions: 3807

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

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

%C From _Jose Aranda_, Aug 04 2024: (Start)

%C About those unknown terms:

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

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

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

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

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

%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

%Y Cf. A081120, A081121, A179163-A179174.

%K nonn,hard,more

%O 0,1

%A _Artur Jasinski_, Jun 30 2010

%E Edited and a(7), a(11), a(13) added by _Ray Chandler_, Jul 11 2010