A392549 - OEIS

A392549

a(n) is the least positive integer k such that Mordell's equation y^2 = x^3 + k^2 has exactly n integer solutions with y >= 0.

%I #24 Jan 29 2026 17:05:02

%S 2,11,1,6,3,10,80,62,63,210,55,840,15,440,120,960,3240,561,2415,510,

%T 665,19320,1155,5320,148785,31185,9240,665000,73920,143640,249480,

%U 379610,54285

%N a(n) is the least positive integer k such that Mordell's equation y^2 = x^3 + k^2 has exactly n integer solutions with y >= 0.

%C a(34) > 10^6 if it exists, a(37) = 255255, a(38) = 434280, a(42) = 794310, a(64) = 451605.

%e a(6) = 10 because 10 is the least k such that y^2 = x^3 + k^2 has 6 integral solutions with nonnegative y: {{-4, 6}, {0, 10}, {5, 15}, {20, 90}, {24, 118}, {2660, 137190}}.

%t f[k_]:=(m=Range[0,71];

%t xmax=50000; (* This xmax only gives correct results up to n = 12, as n=13 has a larger solution {720114, 611085363} requiring a larger xmax *)

%t m72k=Intersection[Union[Mod[m^2-k^2,72]],Union[Mod[m^3,72]]];

%t s=Select[Range[-Floor[k^(2/3)],xmax],MemberQ[m72k,Mod[#^3,72]]&];t=Select[s,IntegerQ[Sqrt[#^3+k^2]]&];{k,Length@t});v=Table[f[k],{k,100}];

%t a=Table[SelectFirst[v,#[[2]]==n&],{n,9}][[All,1]]

%Y Cf. A081119, A134108, A392548.

%K nonn,more

%O 1,1

%A _Zhining Yang_, Jan 15 2026