A134042 -id:A134042 - 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

A134043 Complete list of solutions to y^2 = x^3 + 113; sequence gives y values.

Original entry on oeis.org

7, 11, 25, 38, 133, 8669

Offset: 1

Artur Jasinski, Oct 03 2007

For corresponding x values see A134043.

			a(1)^2 = 7^2 = 49 = A134042(1)^3 + 113 = -64 + 113.
a(2)^2 = 11^2 = 121 = A134042(2)^3 + 113 = 8 + 113.
a(3)^2 = 25^2 = 625 = A134042(3)^3 + 113 = 512 + 113.
a(4)^2 = 38^2 = 1444 = A134042(4)^3 + 113 = 1331+ 113.
a(5)^2 = 133^2 = 17689 = A134042(5)^3 + 113 = 17576 + 113.
a(6)^2 = 8669^2 = 75151561 = A134042(6)^3 + 113 = 75151448 + 113.

Cf. A134042, A029727, A080761.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 113])) ]); /* adapted from A029727 */

(Program does not produce first two terms) a = {}; Do[k = n^2 - (Floor[n^(2/3)])^3; If[(k > 112) && (k < 113), AppendTo[a, n]], {n, 1, 100000}]; a

Edited and corrected by Klaus Brockhaus, Oct 04 2007

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

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

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134106.

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

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, -207]).integral_points()] # Seiichi Manyama, Aug 25 2019

A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

Original entry on oeis.org

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151

Offset: 1

Cino Hilliard, Mar 10 2003

nonn

From Artur Jasinski, Oct 03 2007: (Start)
Some numbers have multiple partitions:
  8 = 4^2 - 8^3 = 312^2 - 46^3,
  9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)
This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

			8 is in the sequence since 3^2 = 1^3 + 8.

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]
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k.

Complement of A080762.
Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
Cf. A029727, A029728, A134042, A134043.

With[{nn=100},Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn],2],#>0&]],nn]] (* Harvey P. Dale, Jul 10 2012 *)

diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),print1(p" "); break) ) ) }

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007

A134074 Complete list of solutions to y^2 = x^3 + 73; sequence gives x values.

Original entry on oeis.org

-4, 2, 3, 6, 72, 356

Offset: 1

Klaus Brockhaus, Oct 07 2007

For corresponding y values and examples see A134073.

Cf. A134073, A029728, A134042.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 73])) ]); /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 73]).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

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

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

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

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A134043 Complete list of solutions to y^2 = x^3 + 113; sequence gives y values.

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

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

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A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

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A134074 Complete list of solutions to y^2 = x^3 + 73; 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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A134167 Complete list of solutions to y^2 = x^3 + 1025; sequence gives x values.

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