A356709 -id:A356709 - 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

A356706 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3.

Original entry on oeis.org

5, 7, 1, 5, 1, 1, 3, 9, 5, 5, 3, 1, 1, 5, 1, 5, 1, 7, 1, 1, 3, 3, 3, 1, 5, 3, 1, 5, 1, 1, 1, 9, 5, 3, 3, 5, 5, 3, 1, 5, 1, 1, 1, 3, 1, 3, 1, 1, 5, 7, 1, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 1, 3, 5, 17, 1, 1, 1, 1, 5, 3, 9, 1, 3, 1, 1, 1, 9, 1, 1, 5, 1, 1, 5, 1, 3, 1, 5, 1, 5, 5, 3, 1, 1, 3, 1, 1

Offset: 1

Jianing Song, Aug 23 2022

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

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A081119, A356707, A356708.

Indices of 1, 3, 5, and 7: A356709, A356710, A356711, A356712.

[len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True)) for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022

a(n) = A081119(n^3).

a(21) corrected and a(22)-a(60) from Lucas A. Brown, Sep 03 2022

Terms a(61) onward from Max Alekseyev, Jun 01 2023

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

A356707 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y positive.

Original entry on oeis.org

2, 3, 0, 2, 0, 0, 1, 4, 2, 2, 1, 0, 0, 2, 0, 2, 0, 3, 0, 0, 1, 1, 1, 0, 2, 1, 0, 2, 0, 0, 0, 4, 2, 1, 1, 2, 2, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 2, 3, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 1, 2, 8, 0, 0, 0, 0, 2, 1, 4, 0, 1, 0, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 0, 1, 0, 0, 3, 1, 2

Offset: 1

Jianing Song, Aug 23 2022

Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3 apart from the trivial solution (-n,0).

			a(2) = 3 because the solutions to y^2 = x^3 + 2^3 with y > 0 are (1,3), (2,4), and (46,312).

Cf. A081119, A356706, A356708.

Indices of 0, 1, 2, and 3: A356709, A356710, A356711, A356712.

[(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))-1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 03 2022

a(n) = (A081119(n^3)-1)/2 = (A356706(n)-1)/2 = A356706(n) - A356708(n).

Offset and a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 03 2022

a(61)-a(100) from Max Alekseyev, Jun 01 2023

A356708 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y nonnegative.

Original entry on oeis.org

3, 4, 1, 3, 1, 1, 2, 5, 3, 3, 2, 1, 1, 3, 1, 3, 1, 4, 1, 1, 2, 2, 2, 1, 3, 2, 1, 3, 1, 1, 1, 5, 3, 2, 2, 3, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 3, 4, 1, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 2, 3, 9, 1, 1, 1, 1, 3, 2, 5, 1, 2, 1, 1, 1, 5, 1, 1, 3, 1, 1, 3, 1, 2, 1, 3, 1, 3, 3, 2, 1, 1, 2, 1, 1, 4, 2, 3

Offset: 1

Jianing Song, Aug 23 2022

Equivalently, number of different values of x in the integral solutions to the Mordell's equation y^2 = x^3 + n^3.

			a(2) = 4 because the solutions to y^2 = x^3 + 2^3 with y >= 0 are (-2,0), (1,3), (2,4), and (46,312).

Cf. A081119, A134108, A356706, A356707.

Indices of 1, 2, 3, and 4: A356709, A356710, A356711, A356712.

[(len(EllipticCurve(QQ, [0, n^3]).integral_points(both_signs=True))+1)/2 for n in range(1, 61)] # Lucas A. Brown, Sep 04 2022

a(n) = (A081119(n^3)+1)/2 = A134108(n^3) = (A356706(n)+1)/2 = A356707(n)+1.

a(21) corrected and a(22)-a(60) by Lucas A. Brown, Sep 04 2022

a(61)-a(100) 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.

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).

A228948 Numbers n such that n^3 + k^2 = m^3 for some k>0, m>0.

Original entry on oeis.org

6, 7, 11, 23, 24, 26, 28, 31, 38, 42, 44, 47, 54, 55, 61, 63, 84, 91, 92, 95, 96, 99, 104, 110, 111, 112, 118, 119, 124, 138

Offset: 1

M. F. Hasler, Oct 05 2013

nonn

Cube root of perfect cubes in A087285 or in A229618 are in the present sequence, but this does not yield all terms, because these sequences require k^2 to be the largest square < m^3.

Numbers k such that Mordell's equation y^2 = x^3 - k^3 has more than 1 integral solution. (Note that it is necessary that x is positive.) In other words, numbers k such that Mordell's equation y^2 = x^3 - k^3 has solutions other than the trivial solution (k,0). - Jianing Song, Sep 24 2022

			6 is a term since the equation y^2 = x^3 - 6^3 has 5 solutions (6,0), (10,+-28), and (33,+-189). - _Jianing Song_, Sep 24 2022

Cf. A229618, A087285, A087286, A088017, A081121, A081120, A077116, A065733.
Cube root of A179419.
Cf. A356709, A356720. Complement of A356713.

More terms added by Jianing Song, Sep 24 2022 based on A179419.

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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A356706 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3.

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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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A356707 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y positive.

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A356708 Number of integral solutions to Mordell's equation y^2 = x^3 + n^3 with y nonnegative.

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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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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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