A081119 -id:A081119 - cp's OEIS frontend

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.

2, 2, 2, 3, 2, 4, 3, 4, 2, 8, 2, 4, 8, 5, 2, 7, 2, 9, 8, 4, 2, 15, 3, 4, 5, 10, 2, 15, 2, 7, 8, 4, 8, 17, 2, 4, 8, 15, 2, 15, 2, 10, 14, 4, 2, 22, 3, 7, 8, 9, 2, 10, 8, 15, 8, 4, 2, 38, 2, 4, 14, 8, 8, 15, 2, 9, 7, 16, 2, 27, 2, 4, 13, 9, 8, 15, 2, 22, 6, 4, 2, 39, 8, 4, 7, 16, 2, 27, 8, 10

Offset: 3

Ctibor O. Zizka, Sep 01 2023

nonn

The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section.
For a prime number p >= 5, the equation y^2 = p^3 + (p + r)^2 has 2 solutions, r_1 = p*(p - 3)/2 and r_2 = (p + 1)*(p^2 - p - 1)/2.
Factoring the equation y^2 = n^3 + n^2 + 2*r*n + r^2 yields (y+n+r)*(y-n-r) = n^3, which implies y+n+r = d and y-n-r = n^3/d for some divisor d of n^3.  Thus a(n) is the number of divisors d of n^3 such that (d-n^3/d)/2 - n is a nonnegative integer.  This resolves some of Thomas Scheuerle's conjectures. - Robin Visser, Sep 30 2023

			n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47, thus a(6) = 3. The 3 solutions [y, n, n+r] are [21, 6, 15], [29, 6, 25], [55, 6, 53].

Robin Visser, Table of n, a(n) for n = 3..10000
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A081119, A134108, A190300, A196226, A245685, A364248.

a(n) = length(select((x) -> x[1] >= 0 && x[2] >= n, thue(thueinit(x^2-1,1),n^3),1)) \\ Thomas Scheuerle, Sep 03 2023

def a(n):
    num_sols = 0
    for d in Integer(n^3).divisors():
        if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): num_sols += 1
    return num_sols  # Robin Visser, Sep 30 2023

a(p) = 2 for p prime >= 5, see Comments.
From Thomas Scheuerle, Sep 04 2023: (Start)
Conjecture: a(A190300(n)) = 3.
Conjecture: a(A196226(n)) = 4.
Conjecture: a(p^3) = 5 if p is an odd prime.
Conjecture: a(2*p^2) = 7 if p is an odd prime. But there exist other cases too, for example a(3*23) = 7.
Conjecture: a(prime(n)^prime(n)) = A245685(n - 1) - 1. (End)

a(61)-a(92) from Thomas Scheuerle, Sep 01 2023

A373795 a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, 113, 2089, 28279, 975379

Offset: 0

N. J. A. Sloane, Jul 04 2024

a(n) = min{ A031507(n), A031508(n) }.
See A031507 and A031508 for further information.
a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - N. J. A. Sloane, Jul 05 2024

Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

Noam D. Elkies and Zev Klagsbrun, New rank records for elliptic curves having rational torsion, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.

Cf. A031507, A031508, A081119.

A179146 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 2 integral solutions.

Original entry on oeis.org

2, 3, 4, 5, 10, 16, 18, 19, 22, 25, 26, 30, 31, 33, 35, 38, 40, 41, 43, 48, 49, 50, 52, 54, 55, 56, 71, 72, 76, 79, 81, 82, 91, 92, 94, 97, 98, 99, 105, 106, 107, 112, 117, 119, 120, 122, 126, 127, 131, 132, 134, 136, 138, 142, 143, 144, 150, 151, 152, 154, 156, 163, 170

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. A054504, A081119, A179145-A179162.

Edited by Ray Chandler, Jul 11 2010

A179148 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 4 integral solutions.

Original entry on oeis.org

12, 15, 28, 44, 63, 68, 101, 121, 128, 148, 168, 197, 198, 204, 208, 220, 232, 248, 269, 294, 337, 346, 350, 369, 404, 409, 443, 481, 485, 492, 540, 556, 561, 575, 618, 640, 656, 659, 701, 702, 716, 740, 757, 768, 775, 785, 804, 829, 850, 857, 868, 885, 901

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. A054504, A081119, A179145-A179162.

URL typos corrected - R. J. Mathar, Jul 05 2010

Edited by Ray Chandler, Jul 11 2010

A179150 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 6 integral solutions.

Original entry on oeis.org

37, 57, 129, 141, 164, 169, 171, 196, 281, 289, 359, 392, 414, 427, 433, 464, 513, 516, 577, 593, 612, 625, 633, 665, 684, 721, 730, 793, 801, 841, 849, 899, 940, 953, 964, 1001, 1081, 1090, 1153, 1169, 1233, 1252, 1289, 1297, 1380, 1441, 1452, 1457, 1500

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. A054504, A081119, A179145-A179162.

Edited by Ray Chandler, Jul 11 2010

A179152 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 8 integral solutions.

Original entry on oeis.org

24, 36, 65, 80, 89, 108, 145, 161, 233, 260, 353, 377, 441, 449, 505, 521, 528, 537, 649, 681, 737, 745, 784, 792, 1100, 1116, 1224, 1296, 1412, 1513, 1536, 1548, 1585, 1753, 1897, 1961, 2025, 2033, 2185, 2250, 2305, 2404, 2521, 2537, 2745, 2793, 2852, 2913

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. A134223, A054504, A081119, A179145-A179162.

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()))

A356704 a(n) is the least k such that Mordell's equation y^2 = x^3 + k^3 has exactly 2*n+1 integral solutions.

Original entry on oeis.org

3, 7, 1, 2, 8, 329, 217, 506, 65, 260, 585

Offset: 0

Jianing Song, Aug 23 2022

a(n) is the least k such that y^2 = x^3 + k^3 has exactly n solutions with y positive, or exactly n+1 solutions with y nonnegative.

a(n) is the smallest index of 2*n+1 in A356706, of n in A356707, and of n+1 in A356708.

			a(4) = 8 since y^2 = x^3 + 8^3 has exactly 9 solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496), and the number of solutions to y^2 = x^3 + k^3 is not 9 for 0 < k < 8.

Cf. A081119, A081120, A179162, A179175, A356705, A356706, A356707, A356708.

a(n) = A179162(2*n+1)^(1/3).

A356705 a(n) is the least k such that Mordell's equation y^2 = x^3 - k^3 has exactly 2*n+1 integral solutions.

Original entry on oeis.org

1, 11, 6, 38, 7, 63, 416, 2600, 10400, 93600

Offset: 0

Jianing Song, Aug 23 2022

a(n) is the least k such that y^2 = x^3 - k^3 has exactly n solutions with y positive, or exactly n+1 solutions with y nonnegative.

			a(1) = 11 since y^2 = x^3 - 11^3 has exactly 3 solutions (11,0) and (443,+-9324), and the number of solutions to y^2 = x^3 - k^3 is not 3 for 0 < k < 11.
a(2) = 6 since y^2 = x^3 - 6^3 has exactly 5 solutions (6,0), (10,+-28), and (33,+-189), and the number of solutions to y^2 = x^3 - k^3 is not 5 for 0 < k < 6.
a(4) = 7 since y^2 = x^3 - 7^3 has exactly 9 solutions (7,0), (8,+-13), (14,+-49), (28,+-147), and (154,+-1911), and the number of solutions to y^2 = x^3 - k^3 is not 9 for 0 < k < 7.

Cf. A081119, A081120, A179162, A179175, A356704.

a(n) = A179175(2*n+1)^(1/3).

a(7)-a(9) from Jose Aranda, Aug 05 2024

cp's OEIS Frontend

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2.

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A373795 a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

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A179146 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 2 integral solutions.

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A179148 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 4 integral solutions.

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A179150 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 6 integral solutions.

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A179152 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 8 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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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

Extensions

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

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Author

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Crossrefs

Programs

Sage

A356704 a(n) is the least k such that Mordell's equation y^2 = x^3 + k^3 has exactly 2*n+1 integral solutions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A356705 a(n) is the least k such that Mordell's equation y^2 = x^3 - k^3 has exactly 2*n+1 integral solutions.

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A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.