A179147 -id:A179147 - cp's OEIS frontend

A356709 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 1 integral solution.

3, 5, 6, 12, 13, 15, 17, 19, 20, 24, 27, 29, 30, 31, 39, 41, 42, 43, 45, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 66, 67, 68, 69, 73, 75, 76, 77, 79, 80, 82, 83, 85, 87, 89, 93, 94, 96, 97, 101, 102, 103, 106, 107, 108, 109, 111, 113, 115, 116, 117, 118, 119

Offset: 1

Jianing Song, Aug 23 2022

nonn

Numbers k such that Mordell's equation y^2 = x^3 + k^3 has no solution other than the trivial solution (-k,0).

Cube root of A179145.

			3 is a term since the equation y^2 = x^3 + 3^3 has no solution other than (-3,0).

Jianing Song, Table of n, a(n) for n = 1..115 (using the b-file of A356720, which is based on the data from A103254)

Cf. A081119, A179145, A179147, A179149, A179151, A356710, A356711, A356712.
Indices of 1 in A356706, of 0 in A356707, and of 1 in A356708.
Complement of A356720.
Cf. also A356713, A228948.

A356711 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

Original entry on oeis.org

1, 4, 9, 10, 14, 16, 25, 28, 33, 36, 37, 40, 49, 64, 70, 81, 84, 88, 90, 91, 100, 104, 121, 126, 130, 132, 140, 144, 154, 160, 169, 176, 184, 193, 196

Offset: 1

Jianing Song, Aug 23 2022

Cube root of A179149.

Contains all squares: suppose that y^2 = x^3 + t^6, then (y/t^3)^2 = (x/t^2)^3 + 1. The elliptic curve Y^2 = X^3 + 1 has rank 0 and the only rational points on it are (-1,0), (0,+-1), and (2,+-3), so y^2 = x^3 + t^6 has 5 solutions (-t^2,0), (0,+-t^3), and (2*t^2,+-3*t^3).

			1 is a term since the equation y^2 = x^3 + 1^3 has 5 solutions (-1,0), (0,+-1), and (2,+-3).

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356712.

Indices of 5 in A356706, of 2 in A356707, and of 3 in A356708.

a(31)-a(35) from Max Alekseyev, Jun 01 2023

A356710 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 3 integral solutions.

Original entry on oeis.org

7, 11, 21, 22, 23, 26, 34, 35, 38, 44, 46, 63, 71, 74, 86, 92, 95, 99, 110, 122, 129, 136, 152, 155, 158, 170, 175, 177, 183, 189, 190, 198, 201, 203, 207, 211

Offset: 1

Jianing Song, Aug 23 2022

Cube root of A179147.

			7 is a term since the equation y^2 = x^3 + 7^3 has 3 solutions (-7,0) and (21,+-98).

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356711, A356712.

Indices of 3 in A356706, of 1 in A356707, and of 2 in A356708.

a(30)-a(36) from Max Alekseyev, Jun 01 2023

A356712 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 7 integral solutions.

Original entry on oeis.org

2, 18, 50, 56, 57, 98, 112, 114, 148, 162, 224, 228, 273, 280, 330, 336, 338, 364, 448, 504, 513, 578

Offset: 1

Jianing Song, Aug 23 2022

Cube root of A179151.

			2 is a term since the equation y^2 = x^3 + 2^3 has 3 solutions (-2,0), (1,+-3), (2,+-4), and (46,+-312).

Cf. A081119, A179145, A179147, A179149, A179151, A356709, A356710, A356711.

Indices of 7 in A356706, of 3 in A356707, and of 4 in A356708.

A356703 Numbers k such that Mordell elliptic curve y^2 = x^3 + k has a number of integral points that is both odd and > 1.

Original entry on oeis.org

1, 8, 64, 343, 512, 729, 1000, 1331, 2744, 4096, 5832, 9261, 10648, 12167, 15625, 17576, 21952, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 64000, 85184, 97336, 117649, 125000, 175616, 185193, 250047, 262144, 274625, 343000, 357911, 373248, 405224, 474552, 531441, 592704, 636056

Offset: 1

Jianing Song, Aug 23 2022

nonn

Cubes k such that y^2 = x^3 + k has a solution other than (-k^(1/3), 0).

Contains all sixth powers since A179149 does.

			512 is a term since the equation y^2 = x^3 + 512 has 9 integral solutions (-8,0), (-7,+-13), (4,+-24), (8,+-32), and (184,+-2496).

Jianing Song, Table of n, a(n) for n = 1..85

Complement of A179145 among the positive cubes.
Cf. A081119, A179147, A179149, A179151, A179163, A179419.
Cf. also A356709, A356720, A356713, A228948.

a(n) = A356720(n)^3.

A134221 Values n for which integer solutions of Mordell curve y^2=x^3+n have 2 different values of x (or equivalently of nonnegative y).

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, 343, 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

Offset: 1

Artur Jasinski, Oct 14 2007

nonn

Union of A179147 and A179148.

Cf. A134108, A134220, A134222, A134223.

Numbers n such that A134108(n) = Floor((A081119(n)+1)/2) = 2.

Edited and extended by Ray Chandler, Jul 12 2010

cp's OEIS Frontend

A356709 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 1 integral solution.

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A356711 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 5 integral solutions.

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A356710 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 3 integral solutions.

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A356712 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has exactly 7 integral solutions.

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A356703 Numbers k such that Mordell elliptic curve y^2 = x^3 + k has a number of integral points that is both odd and > 1.

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A134221 Values n for which integer solutions of Mordell curve y^2=x^3+n have 2 different values of x (or equivalently of nonnegative y).

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