A179163 -id:A179163 - 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

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.

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

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

A350976 Numbers k such that x^3 - y^2 = k has a unique solution.

Original entry on oeis.org

2, 13, 19, 44, 49, 74, 146, 193, 301, 506, 589, 767, 769, 866, 868

Offset: 1

N. J. A. Sloane, Mar 06 2022

Apparently this is an incomplete version of A179164. (Note that the sign-switched y are counted individually in A179164 and A179163.) - R. J. Mathar, Mar 07 2022

H. Brocard, Query 2312, L'Intermédiaire des Mathématiciens, 10 (1903), 283-284.

LMFDB 13, 19, 44

Cf. A350977.

A179165 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 4 integral solutions.

Original entry on oeis.org

4, 7, 11, 26, 48, 53, 55, 60, 63, 76, 109, 147, 180, 212, 215, 242, 256, 277, 362, 364, 375, 391, 405, 433, 448, 471, 476, 511, 535, 593, 615, 674, 680, 704, 728, 767, 782, 802, 831, 856, 875, 895, 900, 914, 931, 975, 991, 996, 1055, 1096, 1108, 1144, 1152

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

A179166 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 6 integral solutions.

Original entry on oeis.org

28, 39, 47, 100, 104, 135, 152, 174, 191, 200, 244, 424, 440, 459, 732, 755, 804, 888, 984, 1048, 1075, 1084, 1236, 1259, 1287, 1322, 1432, 1503, 1668, 1763, 1792, 1812, 1951, 2160, 2224, 2344, 2367, 2440, 2468, 2496, 2556, 2692, 2695, 2699, 2727, 2799

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

cp's OEIS Frontend

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

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

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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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A179174 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 22 integral solutions.

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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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A350976 Numbers k such that x^3 - y^2 = k has a unique solution.

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A179165 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 4 integral solutions.

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A179166 Numbers n such that Mordell's equation y^2 = x^3 - n has exactly 6 integral solutions.

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