A228948 -id:A228948 - 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.

A356720 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has more than 1 integral solution.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 14, 16, 18, 21, 22, 23, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 44, 46, 49, 50, 56, 57, 63, 64, 65, 70, 71, 72, 74, 78, 81, 84, 86, 88, 90, 91, 92, 95, 98, 99, 100, 104, 105, 110, 112, 114, 121, 122, 126, 128, 129, 130, 132, 136, 140, 144, 148

Offset: 1

Jianing Song, Aug 24 2022

nonn

Numbers k such that Mordell's equation y^2 = x^3 + k^3 has solutions other than the trivial solution (-k,0).
Different from A103254, which lists k such that Mordell's equation y^2 = x^3 + k^3 has solutions with positive x (or equivalently, with nonnegative x). 71, 74, and 155 are here but not in A103254.
Cube root of A356703.
Contains all squares since A356711 does.

			71 is a term since the equation y^2 = x^3 + 71^3 has 3 solutions (-71,0) and (-23,+-588).
74 is a term since the equation y^2 = x^3 + 74^3 has 3 solutions (-74,0) and (-47,+-549).
155 is a term since the equation y^2 = x^3 + 155^3 has 3 solutions (-155,0) and (-31,+-1922).

Jianing Song, Table of n, a(n) for n = 1..85 (based on the data from A103254)

Cf. A081119, A356703, A356713, A228948, A103254. Complement of A356709.

Cf. also A356710, A356711, A356712.

A356713 Numbers k such that Mordell's equation y^2 = x^3 - k^3 has exactly 1 integral solution.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 45, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

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 A179163.
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), so y^2 = x^3 - t^6 has only one solution (t^2,0).

Jianing Song, Table of n, a(n) for n = 1..108 (using data from A179149)

Cf. A081120, A179163, A356709, A356720. Complement of A228948.

1 is a term since the equation y^2 = x^3 - 1^3 has no solution other than (1,0).

A179419 Numbers n such that Mordell elliptic curve y^2=x^3-n has a number of integral points that is both odd and > 1.

Original entry on oeis.org

216, 343, 1331, 12167, 13824, 17576, 21952, 29791, 54872, 74088, 85184, 103823, 157464, 166375, 226981, 250047, 592704, 753571, 778688, 857375, 884736, 970299, 1124864, 1331000, 1367631, 1404928, 1643032, 1685159, 1906624, 2628072

Offset: 1

Artur Jasinski, Jul 13 2010

nonn

Also positive cubes not in A179163.
A000578 = Union({0}, A179163, A179419).
Mordell curve y^2=x^3-n always has at least one integral solution if n is a cube, say n=k^3, (x,y)=(k,0). If there are additional solutions, they will exist in pairs - (x,y) and (x,-y). Thus the number of solutions can be odd iff n is a cube.

Cf. A000578, A179163. Cube of A228948.

Edited and extended by Ray Chandler, Jul 14 2010

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.

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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A356720 Numbers k such that Mordell's equation y^2 = x^3 + k^3 has more than 1 integral solution.

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A356713 Numbers k such that Mordell's equation y^2 = x^3 - k^3 has exactly 1 integral solution.

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A179419 Numbers n such that Mordell elliptic curve y^2=x^3-n has a number of integral points that is both odd and > 1.

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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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