cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A029728 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.

Original entry on oeis.org

-2, -1, 2, 4, 8, 43, 52, 5234
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Comments by Henri Cohen on the proof that the list of solutions is complete: (Start)
This is now completely standard. Cremona's mwrank program tells us that this is an elliptic curve of rank 2 with generators P1=(-2,3) and P2=(4,9).
We now apply the algorithm (essentially due to Tzanakis and de Weger) described in Nigel Smart's book on the algorithmic solution of Diophantine equations: using Sinnou David's bounds on linear forms in elliptic logarithms one finds that if P is an integral point then P=aP1+bP2 for |a| and |b| less than a huge bound B (typically 10^100, sometimes more, I didn't do the computation here), but the main point is that B is completely explicit. One then uses the LLL algorithm: this is crucial.
A first application reduces the bound to 200, say, then a second application to 20 and sometimes a third to 12 (at this point it is not necessary). Then a very small search gives all the possible integer points. (End)

References

  • L. J. Mordell, Diophantine Equations, Ac. Press, p. 246.
  • T. Nagell, Einige Gleichungen von der Form ay^2+by+c=dx^3, Vid. Akad. Skrifter Oslo, Nr. 7 (1930).
  • Silverman, Joseph H. and John Tate, Rational Points on Elliptic Curves. New York: Springer-Verlag, 1992.

Crossrefs

Cf. A029727 (y values).
x values of solutions to y^2 = x^3 + a*x + b;
A134107 (a= 0, b=-207),
A134074 (a= 0, b= 73),
A134042 (a= 0, b= 113),
A134103 (a= 0, b= 225),
A134105 (a= 0, b= 297),
A134167 (a= 0, b=1025),
A316456 (a=-7, b= 10),
A309071 (a=20, b= 0).

Programs

  • Magma
    Sort([ p[1] : p in IntegralPoints(EllipticCurve([0,17])) ]); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Mathematica
    ok[x_] := Reduce[y>0 && y^2 == x^3 + 17, y, Integers] =!= False; Select[Table[x, {x, -2, 10000}], ok ] (* Jean-François Alcover, Sep 07 2011 *)
  • SageMath
    [i[0] for i in EllipticCurve([0, 17]).integral_points()] # Seiichi Manyama, Aug 25 2019

A134166 Complete list of solutions to y^2 = x^3 + 1025; sequence gives y values.

Original entry on oeis.org

5, 30, 31, 32, 33, 45, 95, 255, 355, 513, 1930, 2139, 9419, 27905, 218796, 227805
Offset: 1

Views

Author

Klaus Brockhaus, Oct 11 2007

Keywords

Comments

For corresponding x values see A134167.

Examples

			a(1)^2 = 5^2 = 25 = A134167(1)^3 + 1025 = -1000 + 1025.
a(2)^2 = 30^2 = 900 = A134167(2)^3 + 1025 = -125 + 1025.
a(3)^2 = 31^2 = 961 = A134167(3)^3 + 1025 = -64 + 1025.
a(4)^2 = 32^2 = 1024 = A134167(4)^3 + 1025 = -1 + 1025.
a(5)^2 = 33^2 = 1089 = A134167(5)^3 + 1025 = 64 + 1025.
a(6)^2 = 45^2 = 2025 = A134167(6)^3 + 1025 = 1000 + 1025.
a(7)^2 = 95^2 = 9025 = A134167(7)^3 + 1025 = 8000 + 1025.
a(8)^2 = 255^2 = 65025 = A134167(8)^3 + 1025 = 64000 + 1025.
a(9)^2 = 355^2 = 126025 = A134167(9)^3 + 1025 = 125000 + 1025.
a(10)^2 = 513^2 = 263169 = A134167(10)^3 + 1025 = 262144 + 1025.
a(11)^2 = 1930^2 = 3724900 = A134167(11)^3 + 1025 = 3723875 + 1025.
a(12)^2 = 2139^2 = 4575321 = A134167(12)^3 + 1025 = 4574296 + 1025.
a(13)^2 = 9419^2 = 88717561 = A134167(13)^3 + 1025 = 88716536 + 1025.
a(14)^2 = 27905^2 = 778689025 = A134167(14)^3 + 1025 = 778688000 + 1025.
a(15)^2 = 218796^2 = 47871689616 = A134167(15)^3 + 1025 = 47871688591 + 1025.
a(16)^2 = 227805^2 = 51895118025 = A134167(16)^3 + 1025 = 51895117000 + 1025.

Crossrefs

Cf. A134167, A029727, A134043, A134073, A134102, A134104, A134106.

Programs

  • Magma
    { x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029727 */
  • Mathematica
    Select[Table[Sqrt[1025+n^3],{n,-10,20000}],IntegerQ] (* Harvey P. Dale, Jan 21 2023 *)
Showing 1-2 of 2 results.

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