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.

Showing 1-5 of 5 results.

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

A134105 Complete list of solutions to y^2 = x^3 + 297; sequence gives x values.

Original entry on oeis.org

-6, -2, 3, 4, 12, 34, 48, 1362, 93844
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding y values and examples see A134104.
The parameter -297 of the curve corresponds to A200218(1). a(9)=A200216(1). - Artur Jasinski, Nov 29 2011

Crossrefs

Cf. A134104, A134103, A134107, A029728, A134042, A134074.

Programs

  • Magma
    Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029728 */
  • Mathematica
    sol[x_] := Solve[y > 0 && x^3 - y^2 == -297, y, Integers];
Reap[For[x = 1, x < 10^5, x++, sx = sol[x]; If[sx != {}, xy = {x, y} /. sx[[1]]; Print[xy]; Sow[xy]]; sx = sol[-x]; If[sx != {}, xy = {-x, y} /. sx[[1]]; Print[xy]; Sow[xy]]]][[2, 1]][[All, 1]] // Sort (* Jean-François Alcover, Feb 07 2020 *)
  • SageMath
    [i[0] for i in EllipticCurve([0, 297]).integral_points()] # Seiichi Manyama, Aug 26 2019

A134107 Complete list of solutions to y^2 = x^3 - 207; sequence gives x values.

Original entry on oeis.org

6, 12, 18, 31, 312, 331, 367806
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding y values and examples see A134106.

Crossrefs

Cf. A134106, A134103, A134105, A029728, A134042, A134074.

Programs

  • Magma
    Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029728 */
  • SageMath
    [i[0] for i in EllipticCurve([0, -207]).integral_points()] # Seiichi Manyama, Aug 25 2019

A134102 Complete list of solutions to y^2 = x^3 + 225; sequence gives y values.

Original entry on oeis.org

3, 10, 15, 17, 21, 35, 60, 165, 465, 2415, 6159, 6576, 611085363
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding x values see A134103.

Examples

			a(1)^2 = 3^2 = 9 = A134103(1)^3 + 225 = -216 + 225.
a(2)^2 = 10^2 = 100 = A134103(2)^3 + 225 = -125 + 225.
a(3)^2 = 15^2 = 225 = A134103(3)^3 + 225 = 0 + 225.
a(4)^2 = 17^2 = 289 = A134103(4)^3 + 225 = 64 + 225.
a(5)^2 = 21^2 = 441 = A134103(5)^3 + 225 = 216 + 225.
a(6)^2 = 35^2 = 1225 = A134103(6)^3 + 225 = 1000 + 225.
a(7)^2 = 60^2 = 3600 = A134103(7)^3 + 225 = 3375 + 225.
a(8)^2 = 165^2 = 27225 = A134103(8)^3 + 225 = 27000 + 225.
a(9)^2 = 465^2 = 216225 = A134103(9)^3 + 225 = 216000 + 225.
a(10)^2 = 2415^2 = 5832225 = A134103(10)^3 + 225 = 5832000 + 225.
a(11)^2 = 6159^2 = 37933281 = A134103(11)^3 + 225 = 37933056 + 225.
a(12)^2 = 6576^2 = 43243776 = A134103(12)^3 + 225 = 43243551 + 225.
a(13)^2 = 611085363^2 = 373425320872841769 = A134103(13)^3 + 225 = 373425320872841544 + 225.

Crossrefs

Cf. A134103, A134104, A134106, A029727, A134043, A134073.

Programs

  • Magma
    { x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029727 */
  • Mathematica
    Select[Table[Sqrt[x^3+225],{x,-6,721000}],IntegerQ] (* Harvey P. Dale, Dec 25 2022 *)

A134167 Complete list of solutions to y^2 = x^3 + 1025; sequence gives x values.

Original entry on oeis.org

-10, -5, -4, -1, 4, 10, 20, 40, 50, 64, 155, 166, 446, 920, 3631, 3730
Offset: 1

Views

Author

Klaus Brockhaus, Oct 11 2007

Keywords

Comments

For corresponding y values and examples see A134166.

Crossrefs

Cf. A134166, A029728, A134042, A134074, A134103, A134105, A134107.

Programs

  • Magma
    { x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029728 */
  • SageMath
    [i[0] for i in EllipticCurve([0, 1025]).integral_points()] # Seiichi Manyama, Aug 26 2019
Showing 1-5 of 5 results.

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