A393416 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.

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

Offset

1,2

Comments

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

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

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.

Links

Table of n, a(n) for n=1..8.

Eric Weisstein's World of Mathematics, Mordell Curve.

Xiaoyang Zhang, 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).

Example

57 is in this sequence because 57 is a squarefree number (A005117) and:
  y^2 = x^3 + 57^0 has 3 solutions: {{-1, 0}, {0, -1}, {2, -3}}
  y^2 = x^3 + 57^1 has 3 solutions: {{-2, 7}, {4, 11}, {7, 20}}
  y^2 = x^3 + 57^2 has 3 solutions: {{-12, -39}, {0, 57}, {76, -665}}
  y^2 = x^3 + 57^3 has 4 solutions: {{-57, 0}, {-38, 361}, {112, -1261}, {456, -9747}}
  y^2 = x^3 + 57^4 has 4 solutions: {{-126, 2925}, {0, 3249}, {270, 5499}, {2527, 127072}}
  y^2 = x^3 + 57^5 has 1 solution: {{52684, -12092581}}.

Mathematica

sMod72=Union[Mod[Range[0, 71]^2, 72]];
cMod72=Union[Mod[Range[0, 71]^3, 72]];
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];
m72k=Intersection[aSet, cMod72];
allowed=Table[MemberQ[m72k, cModList[[i+1]]], {i, 0, 71}];
Catch[Do[If[allowed[[Mod[x, 72]+1]], y2=x^3+k^r;
If[IntegerQ[Sqrt[y2]], Throw[{r, k, {x}}]]], {x, low, xmax}]; {r, k, {}}]]
g[k_]:=Catch[Scan[If[f[k, #][[3]]==={}, Throw[False]]&, {1, 5, 2, 4}]; True]
a=Select[Range[60], SquareFreeQ[#]&&g[#]&]

Crossrefs

Cf. A005117, A054504, A081119.

Sequence in context: A307344 A330570 A369305 * A065198 A033866 A023109

Adjacent sequences: A393413 A393414 A393415 * A393417 A393418 A393419

Keyword

nonn,more

Author

Zhining Yang, Feb 14 2026

Status

approved