A134107 -id:A134107 - cp's OEIS frontend

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

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

Offset: 1

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)

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.

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).

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0,17])) ]); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

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 *)

[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

Klaus Brockhaus, Oct 08 2007

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

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

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029728 */

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 *)

[i[0] for i in EllipticCurve([0, 297]).integral_points()] # Seiichi Manyama, Aug 26 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

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134102.

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

{ x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 225]).integral_points()] # Seiichi Manyama, Aug 26 2019

A134106 Complete list of solutions to y^2 = x^3 - 207; sequence gives y values.

Original entry on oeis.org

3, 39, 75, 172, 5511, 6022, 223063347

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding x values see A134107.

			a(1)^2 = 3^2 = 9 = A134107(1)^3 - 207 = 216 - 207.
a(2)^2 = 39^2 = 1521 = A134107(2)^3 - 207 = 1728 - 207.
a(3)^2 = 75^2 = 5625 = A134107(3)^3 - 207 = 5832 - 207.
a(4)^2 = 172^2 = 29584 = A134107(4)^3 - 207 = 29791 - 207.
a(5)^2 = 5511^2 = 30371121 = A134107(5)^3 - 207 = 30371328 - 207.
a(6)^2 = 6022^2 = 36264484 = A134107(6)^3 - 207 = 36264691 - 207.
a(7)^2 = 223063347^2 = 49757256774842409 = A134107(7)^3 - 207 = 49757256774842616 - 207.

Cf. A134107, A134102, A134104, A029727, A134043, A134073.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029727 */

[x[1] for x in EllipticCurve([0,-207]).integral_points()] # Charles R Greathouse IV, Aug 09 2024

A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3

Offset: 1

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

nonn

a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

			y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.
y^2 = x^3 - 5 has no solutions, hence a(5) = 0.
y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.
y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A081120, A134106, A134107, A134108.

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];

A081120 = Cases[Import["https://oeis.org/A081120/b081120.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081120[[n]]}, If[EvenQ[an], an/2, (an+1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 28 2019 *)

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

Klaus Brockhaus, Oct 11 2007

For corresponding y values and examples see A134166.

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

{ x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 1025]).integral_points()] # Seiichi Manyama, Aug 26 2019

A179174 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 22 integral solutions.

Original entry on oeis.org

3807, 3896, 52784, 129556, 157239, 167600, 185112, 200871, 281439, 314199, 347967, 370647, 399375, 553648, 623872, 720703, 815728, 819775, 856799, 934975, 994816

Offset: 1

Artur Jasinski, Jun 30 2010

Counting (+x,+y) and (+x,-y) iff y != 0.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Jose Aranda, C++ Program

Subsequence of A007412.

Cf. A081120, A081121, A179163-A179175, A134107, A106265.

Edited by Ray Chandler, Jul 11 2010

a(3)-a(21) from Jose Aranda, Aug 10 2024

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

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

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

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A134106 Complete list of solutions to y^2 = x^3 - 207; sequence gives y values.

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A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

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

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A179174 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 22 integral solutions.

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