A356720 -id:A356720 - 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

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

Original entry on oeis.org

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.

A103254 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2.

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, 72, 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, 152, 154, 158, 160, 162, 169, 170, 175, 176, 177, 183, 184, 189, 190, 193, 196, 198, 200

Offset: 1

Cino Hilliard, Mar 20 2005

nonn

A001105 is a subset (excluding 0), since (x, y, z) = (A001105(k), A001105(k), A033430(k)) satisfies x^3 + y^3 = z^2. - R. J. Mathar, Sep 11 2006
A parametric solution: {x,y,z} = {g*(4*e + g)*(4*e^2 + 8*e*g + g^2), 2*g*(4*e + g)*(-2*e^2 +2*e*g + g^2), 3*g^2*(4*e + g)^2*(4*e^2 + 2*e*g + g^2)}, provided (-2*e^2 +2*e*g + g^2) > 0. - James Mc Laughlin, Jan 27 2007
Allowing y = 0 would give the same sequence, since x^3 = z^2 implies x is a square, and all squares are terms since (t^2)^3 + (2*t^2)^3 = (3*t^3)^2. On the other hand, allowing y to be negative would introduce new terms: 71, 74, and 155 would be terms since 71^3 + (-23)^3 = 588^2, 74^3 + (-47)^3 = 549^2, and 155^3 + (-31)^3 = 1922^2. See A356720. - Jianing Song, Aug 24 2022

			x=7, y=21, 7^3 + 21^3 = 98^2. 7 is the 4th term in the list.
Other solutions are (x, y, z)=(1, 2, 3), (4, 8, 24), (7, 21, 98), (9, 18, 81), (10, 65, 525), (11, 37, 228), (14, 70, 588), (16, 32, 192), (21, 7, 98), (22, 26, 168), (23, 1177, 40380), ...

Fritz Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.

See A103255 for another version.

[ k : k in [1..200] | exists{P : P in IntegralPoints(EllipticCurve([0,k^3])) | P[1] gt 0 and P[2] ne 0 } ]; // Geoff Bailey, Jan 28 2007

Recomputed and extended to 48 terms by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using Magma, Jan 28 2007

Terms 104..200 added by Joerg Arndt, Sep 29 2012

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.

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

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

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

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A103254 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2.

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

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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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