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 1-7 of 7 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

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

A134103 Complete list of solutions to y^2 = x^3 + 225; sequence gives x values.

Original entry on oeis.org

-6, -5, 0, 4, 6, 10, 15, 30, 60, 180, 336, 351, 720114
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding y values and examples see A134102.

Crossrefs

Cf. A134102, A134105, A134107, A029728, A134042, A134074.

Programs

  • Magma
    { x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029728 */
  • SageMath
    [i[0] for i in EllipticCurve([0, 225]).integral_points()] # Seiichi Manyama, Aug 26 2019

A134104 Complete list of solutions to y^2 = x^3 + 297; sequence gives y values.

Original entry on oeis.org

9, 17, 18, 19, 45, 199, 333, 50265, 28748141
Offset: 1

Views

Author

Klaus Brockhaus, Oct 08 2007

Keywords

Comments

For corresponding x values see A134105.

Examples

			a(1)^2 = 9^2 = 81 = A134105(1)^3 + 297 = -216 + 297.
a(2)^2 = 17^2 = 289 = A134105(2)^3 + 297 = -8 + 297.
a(3)^2 = 18^2 = 324 = A134105(3)^3 + 297 = 27 + 297.
a(4)^2 = 19^2 = 361 = A134105(4)^3 + 297 = 64 + 297.
a(5)^2 = 45^2 = 2025 = A134105(5)^3 + 297 = 1728 + 297.
a(6)^2 = 199^2 = 39601 = A134105(6)^3 + 297 = 39304 + 297.
a(7)^2 = 333^2 = 110889 = A134105(7)^3 + 297 = 110592 + 297.
a(8)^2 = 50265^2 = 2526570225 = A134105(8)^3 + 297 = 2526569928 + 297.
a(9)^2 = 28748141^2 = 826455610955881 = A134105(9)^3 + 297 = 826455610955584 + 297.

Crossrefs

Cf. A134105, A134102, A134106, A029727, A134043, A134073.

Programs

  • Magma
    Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029727 */
  • 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, 2]] // Sort (* Jean-François Alcover, Feb 07 2020 *)

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

A201268 Distances d=x^3-y^2 for primary extremal points {x,y} of Mordell elliptic curves with quadratic extensions over rationals.

Original entry on oeis.org

52488, 15336, -20088, 219375, -293625, -474552, 1367631, -297, 100872, -105624, 6021000, -6615000, 40608000, -45360000, -423360000, 69641775, -72560097, 110160000, -114912000, -1216512, 1418946687, -1507379625, 1450230912, -1533752064, 2143550952, 4566375
Offset: 1

Views

Author

Artur Jasinski, Nov 29 2011

Keywords

Comments

For successive x coordinates see A201047.
For successive y coordinates see A201269.
One elliptic curve with particular d can contain a finite number of extremal points.
Theorem (*Artur Jasinski*):
One elliptic curve cannot contain more than 1 extremal primary point with quadratic extension over rationals.
Consequence of this theorem is that any number in this sequence can't appear more than 1 time.
Conjecture (*Artur Jasinski*):
One elliptic curve cannot contain more than 1 point with quadratic extension over rationals.
Mordell elliptic curves contained points with extensions which are roots of polynomials : 2 degree (with Galois 2T1), 4 degree (with Galois 4T3) and 6 degree (with not soluble Galois PGL(2,5) ). Order of minimal polynomial of any extension have to divided number 12. Theoretically points can exist which are roots of polynomial of 3 degree but any such point isn't known yet.
Particular elliptic curves x^3-y^2=d can contain more than one extremal point e.g. curve x^3-y^2=-297=a(8) contained 3 of such points with coordinates x={48, 1362, 93844}={A134105(7),A134105(8),A134105(9)}.
Conjecture (*Artur Jasinski*): Extremal points are k-th successive points with maximal coordinates x.

Crossrefs

Cf. A200218, A201047, A201269.

Formula

a(n) = (A201047(n))^3-(A201269(n))^2.

A200565 Integral x solutions of elliptic curve x^3-y^2 = 54814765 = A200218(2).

Original entry on oeis.org

819, 5256, 838044, 322001299796379844
Offset: 1

Views

Author

Artur Jasinski, Nov 19 2011

Keywords

Comments

a(4)=A200216(2).

Crossrefs

Cf. A200216, A200217, A200218, A134105.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.