A081120 -id:A081120 - cp's OEIS frontend

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

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 *)

A228948 Numbers n such that n^3 + k^2 = m^3 for some k>0, m>0.

Original entry on oeis.org

6, 7, 11, 23, 24, 26, 28, 31, 38, 42, 44, 47, 54, 55, 61, 63, 84, 91, 92, 95, 96, 99, 104, 110, 111, 112, 118, 119, 124, 138

Offset: 1

M. F. Hasler, Oct 05 2013

nonn

Cube root of perfect cubes in A087285 or in A229618 are in the present sequence, but this does not yield all terms, because these sequences require k^2 to be the largest square < m^3.

Numbers k such that Mordell's equation y^2 = x^3 - k^3 has more than 1 integral solution. (Note that it is necessary that x is positive.) In other words, numbers k such that Mordell's equation y^2 = x^3 - k^3 has solutions other than the trivial solution (k,0). - Jianing Song, Sep 24 2022

			6 is a term since the equation y^2 = x^3 - 6^3 has 5 solutions (6,0), (10,+-28), and (33,+-189). - _Jianing Song_, Sep 24 2022

Cf. A229618, A087285, A087286, A088017, A081121, A081120, A077116, A065733.
Cube root of A179419.
Cf. A356709, A356720. Complement of A356713.

More terms added by Jianing Song, Sep 24 2022 based on A179419.

A356699 Numbers k such that Mordell's equation y^2 = x^3 + k has a record number of integral solutions.

Original entry on oeis.org

1, 8, 9, 17, 225, 1025

Offset: 1

Jianing Song, Aug 23 2022

Indices of records in A081119. For the record values see A356701.

			1025 is a term since the equation y^2 = x^3 + 1025 has 32 integral solutions, and the number of solutions to y^2 = x^3 + k is less than 32 for 0 < k < 1025.

Cf. A081119, A081120, A356700, A356701, A356702.

A356700 Numbers k such that Mordell's equation y^2 = x^3 - k has a record number of integral solutions.

Original entry on oeis.org

1, 2, 4, 28, 116, 207, 431, 2351, 3807

Offset: 1

Jianing Song, Aug 23 2022

Indices of records in A081120. For the record values see A356702.

			207 is a term since the equation y^2 = x^3 + 207 has 14 integral solutions, and the number of solutions to y^2 = x^3 - k is less than 14 for 0 < k < 207.

Cf. A081119, A081120, A356699, A356701, A356702.

A356701 Records values in A081119.

Original entry on oeis.org

5, 7, 10, 16, 26, 32

Offset: 1

Jianing Song, Aug 23 2022

			a(6) = 32 since A356699(6) = 1025, and the equation y^2 = x^3 + 1025 has 32 integral solutions.

Cf. A081119, A081120, A356699, A356700, A356702.

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

A154332 Least positive integer m such that A087285(n) = A154333(m) = m^3 - next smaller square.

Original entry on oeis.org

3, 2, 32, 15, 17, 4, 7, 6, 35, 8, 11, 10, 14, 21, 12, 28, 65, 9, 56, 18, 136, 568, 23, 99, 101, 20, 13, 27, 34, 30, 143, 145, 38, 16, 19, 47, 195, 91, 197, 175, 26, 51, 59, 799, 69, 62, 163, 255, 257, 66, 31, 717, 2904, 33, 377, 79, 323, 325, 25

Offset: 1

M. F. Hasler, Jan 07 2009

nonn

The terms of this sequence constitute a "proof" for the terms listed in A087285. To prove that a number is NOT in A087285, one can check the finite number (A081120) of solutions to the corresponding Mordell equation, cf. references in A081121.

A154332(n) = { local(m); until(m++^3-sqrtint(m^3-1)^2==A087285[n],); m }

A087285(n) = A154333(a(n)) = a(n)^3 - [sqrt(a(n)^3 - 1)]^2 = A000578(a(n)) - A048760(a(n)^3-1).

A179164 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 2 integral solutions.

Original entry on oeis.org

2, 13, 15, 18, 19, 20, 23, 25, 35, 40, 44, 45, 49, 54, 56, 61, 67, 71, 72, 74, 79, 81, 83, 87, 89, 95, 106, 107, 112, 118, 121, 124, 126, 127, 128, 139, 143, 146, 148, 150, 151, 153, 155, 159, 167, 170, 172, 175, 184, 186, 188, 193, 199, 222, 223, 233, 235, 236, 239

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

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]

Cf. A081120, A081121, A179163-A179175.

Edited by Ray Chandler, Jul 11 2010

A258928 a(n) = number of integral points on the elliptic curve y^2 = x^3 - (n^2)*x + 1, considering only nonnegative values of y.

Original entry on oeis.org

3, 6, 11, 9, 15, 13, 14, 17, 26, 12, 12, 11, 12, 19, 20, 11, 19, 36, 12, 17, 16, 11, 19, 16, 15, 27, 17, 17, 18, 16, 12, 15, 17, 11, 12, 11, 28, 16, 12, 11, 15, 24, 27, 11, 17, 12, 26, 15, 17, 15, 12, 15, 17, 27, 12, 14, 16, 15, 16, 24, 12, 41, 17, 16, 12, 11, 17, 16, 16, 15, 23, 15, 16, 20, 15

Offset: 0

Morris Neene, Jun 14 2015

nonn

For n>3, the number of integral points on y = x^3 - (n^2)*x + 1 is at least 11. These 11 points correspond to the solutions x = {-1, 0, n, -n, n + 2, -n + 2, n^2 - 1, n^2 - 2n + 2, n^2 + 2n + 2, n^4 + 2n, n^4 - 2n}.

			a(0) = 3 because the integer points on y^2 = x^3 + 1 are (-1, 0), (0, 1), and (2, 3).

Robert Israel, Table of n, a(n) for n = 0..209

Cf. A081119, A081120, A259191.

def f(n):
  R. = QQ[]
  E = EllipticCurve(y^2 - x^3 + n^2*x - 1)
  return len(E.integral_points(both_signs=false))
[f(x) for x in range(40)]  # Robert Israel, Apr 23 2021

More terms from Robert Israel, Apr 23 2021

A259191 Number of integral solutions to y^2 = x^3 + n*x^2 + n (with y nonnegative).

Original entry on oeis.org

3, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 1, 8, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 6, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 6, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0

Offset: 1

Morris Neene, Jun 20 2015

nonn

If n is square there are at least two solutions, corresponding to x = 0 and x = -n. If n = 2^(2k) there are at least three solutions, corresponding to x = 0, x = -2^(2k), and x = 2^(6k-2) + 2^(2k). If n = 2k^2 + 2k, there is at least one solution, corresponding to x = 1.

Cf. A081119, A081120.

for i in range(1,31):
    E=EllipticCurve([0,i,0,0,i])
    print(len(E.integral_points()))

cp's OEIS Frontend

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

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Magma

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A228948 Numbers n such that n^3 + k^2 = m^3 for some k>0, m>0.

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A356699 Numbers k such that Mordell's equation y^2 = x^3 + k has a record number of integral solutions.

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A356700 Numbers k such that Mordell's equation y^2 = x^3 - k has a record number of integral solutions.

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A356701 Records values in A081119.

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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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A154332 Least positive integer m such that A087285(n) = A154333(m) = m^3 - next smaller square.

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PARI

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A179164 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 2 integral solutions.

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A258928 a(n) = number of integral points on the elliptic curve y^2 = x^3 - (n^2)*x + 1, considering only nonnegative values of y.

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Sage

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A259191 Number of integral solutions to y^2 = x^3 + n*x^2 + n (with y nonnegative).

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