A029728 -id:A029728 - cp's OEIS frontend

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

-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

A124439 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

The solutions here are listed in the order given by Mordell. See A029728 for a better version (with comments and 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).

Cf. A124451 (y values), A029728.

A124451 Complete list of solutions to y^2 = x^3 + 17; sequence gives y values.

Original entry on oeis.org

4, 3, 5, 9, 23, 282, 375, 378661

Offset: 1

N. J. A. Sloane

The solutions here are listed in the order given by Mordell. See A029728 and A029727 for a better version (with comments and references).

L. J. Mordell, Diophantine Equations, Ac. Press, p. 246.

Cf. A124439 (x values). See A029728 for further comments and references.

A303615 Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.

Original entry on oeis.org

-29, -25, -20, -14, -5, 5, 14, 16, 20, 25, 49, 70, 79, 130, 250, 305, 400, 695, 1555, 1645, 18895

Offset: 1

Tomohiro Yamada, May 29 2018

This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points in this curve.

This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).

			a(6) = 5: 5^3 - 525 * 5 + 10156 = 7656 = 88 * 87.

Roelof J. Stroeker and Benjamin M. M. de Weger, Elliptic binomial diophantine equations, Math. Comp. 68 (1999), 1257-1281.

Cf. A029728 (the complete list of solutions x to y^2=x^3+17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).

A309071 Complete list of solutions to y^2 = x^3 + 20*x; sequence gives x values.

Original entry on oeis.org

0, 4, 5, 720

Offset: 1

Seiichi Manyama, Jul 10 2019

(x,y) = (0,0) is this solution. Consider x > 0. If x is square, x^2 + 20 is square and we get (x,y) = (4,12). If x is not square, x = i^2*j where j is squarefree. j | x^2 + 20, so j is 2,5 or 10. If j = 2 or 10, there is no such (x,y). If j = 5, (y/5)^2 = i^2*(5*i^4 + 4). So 5*i^4 + 4 = k^2. That is k^2 - 5*i^4 = 4. i^2 is a square Fibonacci number. i^2 = 1 or 144, so x = 5 or 720.

			  0^3 + 20*  0 =         0 =     0^2.
  4^3 + 20*  4 =       144 =    12^2.
  5^3 + 20*  5 =       225 =    15^2.
720^3 + 20*720 = 373262400 = 19320^2.

Cf. A029728, A116108, A126203.

for(k=0, 1e5, if(issquare(k*(k^2+20)), print1(k", ")))

[i[0] for i in EllipticCurve([20, 0]).integral_points()] # Seiichi Manyama, Aug 25 2019

A316456 Complete list of solutions to y^2 = x^3 - 7x + 10; sequence gives x values.

Original entry on oeis.org

-3, -2, -1, 1, 2, 3, 5, 9, 13, 31, 41, 67, 302

Offset: 1

Tomohiro Yamada, Jul 04 2018

Bremner and Tzanakis showed that the list of solutions is complete.
The elliptic curve given by this equation has rank 2 over the rationals with generators (1, 2) and (2, 2).
Since there exist two integer points (x, y) and (x, -y) for each x in the sequence (we can easily see that y <> 0 for such an x), this elliptic curve has exactly 26 integer points.

Andrew Bremner and Nicholas Tzanakis, Integer points on y^2 = x^3 - 7x + 10, Math. Comp. 41 (1983), 731-741.
Robin Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, Chapter IV, Exercise 4.18.

Cf. A029728 (y^2 = x^3 + 17), A047694 (y^2 = C(x,0) + C(x,1) + C(x,2) + C(x,3))

EllipticCurve([0,0,0,-7,10]).integral_points()

A317053 Complete list of solutions to y^2 + y = x^3 - 525x + 10156 with y > 0; sequence gives positive y values.

Original entry on oeis.org

31, 74, 76, 87, 112, 121, 319, 562, 679, 1462, 3937, 5312, 7987, 18312, 61312, 66712, 2597287

Offset: 1

Tomohiro Yamada, Jul 20 2018

This equation gives the elliptic curve (W46) studied by Stroeker and de Weger. This curve has rank 3 with generators P1 = (25, 112), P2 = (-20, 112) and P3 = (70, 562). The list gives all integer points with y > 0 in this curve.
Each positive y corresponds to a negative solution -y - 1, so that the sequence gives all y values of solutions.
Some y values corresponds to three solutions. For y = 87, we have x = -25, 5 or 20. For y = 112, we have x = -20, -5 or 25. Any other value of y corresponds to a unique solution.
This equation can be transformed to A000332(n) = A000579(m) by x = (15/2)m^2 - (75/2)m + 25 and y = (225/2)n^2 - (675/2)n + 112. Hence, A000332(n) = A000579(m) (n >= 4, m >= 6) has no integer solutions other than (n, m)= (4, 6) and (10, 10).

			a(1) = 31: (-29)^3 - 525 * (-29) + 10156 = 996 = 31 * 32.

Roelof J. Stroeker and Benjamin M. M. de Weger, Elliptic binomial diophantine equations, Math. Comp. 68 (1999), 1257-1281.

Cf. A303615 (x values)

Cf. A029728 (the complete list of solutions x to y^2 = x^3 + 17), A102461 (the complete list of solutions n to A000217(n) = A027568(m)).

EllipticCurve([0, 0, 1, -525, 10156]).integral_points()

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

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A124439 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.

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A124451 Complete list of solutions to y^2 = x^3 + 17; sequence gives y values.

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A303615 Complete list of solutions to y^2 + y = x^3 - 525x + 10156; sequence gives x values.

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A309071 Complete list of solutions to y^2 = x^3 + 20*x; sequence gives x values.

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PARI

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A316456 Complete list of solutions to y^2 = x^3 - 7x + 10; sequence gives x values.

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SageMath

A317053 Complete list of solutions to y^2 + y = x^3 - 525x + 10156 with y > 0; sequence gives positive y values.

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Programs

SageMath