A179162 -id:A179162 - cp's OEIS frontend

A179145 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 1 integral solution.

27, 125, 216, 1728, 2197, 3375, 4913, 6859, 8000, 13824, 19683, 24389, 27000, 29791, 59319, 68921, 74088, 79507, 91125, 103823, 110592, 132651, 140608, 148877, 157464, 166375, 195112, 205379, 216000, 226981, 238328, 287496, 300763, 314432

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

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)
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, A356709.
Complement of A356703 among the positive cubes.
Cf. also A179163, A179419.

(* Assuming every term is a cube *) xmax = 2000; r[n_] := Reap[ Do[ rpos = Reduce[y^2 == x^3 + n, y, Integers]; If[rpos =!= False, Sow[rpos]]; rneg = Reduce[y^2 == (-x)^3 + n, y, Integers]; If[rneg =!= False, Sow[rneg]], {x, 1, xmax}]]; ok[n_] := Which[ rn = r[n]; rn[[2]] === {}, False, Length[rn[[2]]] > 1, False, ! FreeQ[rn[[2, 1]], Or], False, True, True]; ok[n_ /; !IntegerQ[n^(1/3)]] = False; ok[1]=False; A179145 = Reap[ Do[ If[ok[n], Print[n]; Sow[n]], {n, 1, 320000}]][[2, 1]] (* Jean-François Alcover, Apr 12 2012 *)

a(n) = A356709(n)^3. - Jianing Song, Aug 24 2022

Edited and extended by Ray Chandler, Jul 11 2010

A179175 a(n) = least positive k such that Mordell's equation y^2 = x^3 - k has exactly n integral solutions.

Original entry on oeis.org

3, 1, 2, 1331, 4, 216, 28, 54872, 116, 343, 828, 250047, 496, 71991296, 207

Offset: 0

Artur Jasinski, Jun 30 2010

The status of further terms is:
  15 integral solutions: unknown
  16 integral solutions: 503
  17 integral solutions: unknown
  18 integral solutions: 431
  19 integral solutions: unknown
  20 integral solutions: 2351
  21 integral solutions: unknown
  22 integral solutions: 3807
For least positive k such that equation y^2 = x^3 + k has exactly n integral solutions, see A179162.
If n is odd, then a(n) is perfect cube. [Ray Chandler]
From Jose Aranda, Aug 04 2024: (Start)
About those unknown terms:
a(15) <=    2600^3 = (26* 10^2)^3
a(17) <=   10400^3 = (26* 20^2)^3
a(19) <=   93600^3 = (26* 60^2)^3
a(21) <= 4586400^3 = (26*420^2)^3
The term a(13) = 71991296 = 416^3 = (26*4^2)^3. (End)

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-A179174.

Edited and a(7), a(11), a(13) added by Ray Chandler, Jul 11 2010

A179149 Numbers k such that Mordell's equation y^2 = x^3 + k has exactly 5 integral solutions.

Original entry on oeis.org

1, 64, 729, 1000, 2744, 4096, 15625, 21952, 35937, 46656, 50653, 64000, 117649, 262144, 343000, 531441, 592704, 681472, 729000, 753571, 1000000, 1124864, 1771561, 2000376, 2197000, 2299968, 2744000, 2985984, 3652264, 4096000, 4826809, 5451776, 6229504, 7189057, 7529536

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

Contains all sixth powers: 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). - Jianing Song, Aug 24 2022

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, A356711.

a(n) = A356711(n)^3.

Edited and extended by Ray Chandler, Jul 11 2010

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

A179147 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 3 integral solutions.

Original entry on oeis.org

343, 1331, 9261, 10648, 12167, 17576, 39304, 42875, 54872, 85184, 97336, 250047, 357911, 405224, 636056, 778688, 857375, 970299, 1331000, 1815848, 2146689, 2515456, 3511808, 3723875, 3944312, 4913000, 5359375, 5545233, 6128487, 6751269, 6859000, 7762392, 8120601, 8365427, 8869743, 9393931

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 and extended by Ray Chandler, Jul 11 2010

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

A179151 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 7 integral solutions.

Original entry on oeis.org

8, 5832, 125000, 175616, 185193, 941192, 1404928, 1481544, 3241792, 4251528, 11239424, 11852352, 20346417, 21952000, 35937000, 37933056, 38614472, 48228544, 89915392, 128024064, 135005697, 193100552

Offset: 1

Artur Jasinski, Jun 30 2010

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 and a(3)-a(22) from Ray Chandler, Jul 11 2010

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

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).

cp's OEIS Frontend

A179145 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 1 integral solution.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A179175 a(n) = least positive k such that Mordell's equation y^2 = x^3 - k has exactly n integral solutions.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A179149 Numbers k such that Mordell's equation y^2 = x^3 + k has exactly 5 integral solutions.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A179147 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 3 integral solutions.

Views

Author

Keywords

Links

Crossrefs

Extensions

A179151 Numbers n such that Mordell's equation y^2 = x^3 + n has exactly 7 integral solutions.

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

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