A393416 - OEIS

%I #14 Mar 21 2026 10:02:24

%S 1,10,19,57,185,537,701,710

%N Squarefree numbers n for which the Mordell equation y^2 = x^3 + n^r admits at least one integer solution for each nonnegative integer r.

%C y^2 = x^3 + n^r has 3 integral solutions {{0,-1},{1,0},{3,2}} for all positive integers n when r=0.

%C y^2 = x^3 + n^3 has an integral solution {0,n} for all integers n.

%C y^2 = x^3 + n^(6k+j) <=> (n^{3k}y)^2 = (n^{2k}x)^3+n^{6k+j} with j = {0,1,2,3,4,5}. Thus, if y^2 = x^3 + n^r has an integral solution for r={1,2,4,5}, then it has at least one integral solution for all nonnegative integers r.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>.

%H Xiaoyang Zhang, <a href="https://www.zhihu.com/question/2004649936793380295">The equation y^2 = x^3 + k^s, where k = {10, 19, 57, 185,...} and s is a non-negative integer, always has integer solutions, why? (in Chinese)</a>.

%e 57 is in this sequence because 57 is a squarefree number (A005117) and:

%e   y^2 = x^3 + 57^0 has 3 solutions: {{-1, 0}, {0, -1}, {2, -3}}

%e   y^2 = x^3 + 57^1 has 3 solutions: {{-2, 7}, {4, 11}, {7, 20}}

%e   y^2 = x^3 + 57^2 has 3 solutions: {{-12, -39}, {0, 57}, {76, -665}}

%e   y^2 = x^3 + 57^3 has 4 solutions: {{-57, 0}, {-38, 361}, {112, -1261}, {456, -9747}}

%e   y^2 = x^3 + 57^4 has 4 solutions: {{-126, 2925}, {0, 3249}, {270, 5499}, {2527, 127072}}

%e   y^2 = x^3 + 57^5 has 1 solution: {{52684, -12092581}}.

%t sMod72=Union[Mod[Range[0,71]^2,72]];

%t cMod72=Union[Mod[Range[0,71]^3,72]];

%t cModList=Mod[Range[0,71]^3,72];f[k_,r_]:=Module[{xmax=10^5,low=-Floor[N[k^(r/3)]],c=Mod[k^r,72],aSet,m72k,allowed,x,y2},aSet=Union[Mod[#-c,72]&/@sMod72];

%t m72k=Intersection[aSet,cMod72];

%t allowed=Table[MemberQ[m72k,cModList[[i+1]]],{i,0,71}];

%t Catch[Do[If[allowed[[Mod[x,72]+1]],y2=x^3+k^r;

%t If[IntegerQ[Sqrt[y2]],Throw[{r,k,{x}}]]],{x,low,xmax}];{r,k,{}}]]

%t g[k_]:=Catch[Scan[If[f[k,#][[3]]==={},Throw[False]]&,{1,5,2,4}];True]

%t a=Select[Range[60],SquareFreeQ[#]&&g[#]&]

%Y Cf. A005117, A054504, A081119.

%K nonn,more

%O 1,2

%A _Zhining Yang_, Feb 14 2026