A103254 -id:A103254 - 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.

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.

A103268 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^5.

Original entry on oeis.org

3, 6, 8, 96, 192, 256, 624, 686, 729

Offset: 1

Cino Hilliard, Mar 20 2005

Warning! These terms have not been proved to be correct. There may be missing terms. - N. J. A. Sloane, Jan 05 2007
There are no solutions with (x,y,z) relatively prime. [Bruin]
Trivially, if m^3 + n^3 = z^2, then (z*m)^3 + (z*n)^3 = z^5. So from A103254 we can find many solutions. - James Mc Laughlin, Jan 30 2007
For max(x,y) < 1.1*10^12, there are no more terms < 1458.  Most likely this is true for all x,y. - Chai Wah Wu, Jan 15 2016

			1 + 2^3 = 3^2 so 3^3 + 6^3 = 3^5 and 3 and 6 are terms.
With max(x,y) < 10^4, we have these [x,y,z] triples: [3, 6, 3], [8, 8, 4], [96, 192, 24], [256, 256, 32], [729, 1458, 81], [1944, 1944, 108], [686, 2058, 98], [3696, 4368, 168], [3072, 6144, 192], [8192, 8192, 256], [2508, 8436, 228], ... - _David Broadhurst_, Jan 30 2007
These are variously immediate consequences of 1 + 1 = 2, 1 + 2^3 = 3^2, 1 + 3^3 = 2^2*7 and, much more unexpectedly, 11^3 + 37^3 = 2^4*3^2*19^2. The last example shows that solutions with a common factor are not completely trivial. [Comment based on email from Alf van der Poorten, Feb 15 2007]
624^3 + 14352^3 = 312^5. - _Chai Wah Wu_, Jan 11 2016

F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.

See A114737 for another version.

r[z_] := Reduce[x > 0 && y > 0 && x^3 + y^3 == z^5, {x, y}, Integers];
sols = Reap[Do[rz = r[z]; If[rz =!= False, xyz = {x, y, z} /. {ToRules[rz]}; Print[xyz]; Sow[xyz]], {z, 1, 1000}]][[2, 1]] // Flatten[#, 1]&;
sols[[All, 1]] // Union (* Jean-François Alcover, Oct 18 2019 *)

Corrected by David Broadhurst and others, Jan 30 2007

Term 624 added by Chai Wah Wu, Jan 11 2016

A121980 Positive integers z, without duplication, in x^3+y^3=z^2.

Original entry on oeis.org

1, 3, 4, 8, 13, 24, 27, 28, 32, 49, 64, 81, 98, 104, 108, 125, 147, 168, 181, 189, 192, 216, 224, 228, 256, 312, 343, 351, 361, 375, 388, 392, 500, 507, 512, 525, 549, 588, 648, 671, 676, 729, 756, 784, 832, 847, 864, 1000, 1014, 1029, 1176, 1183, 1225, 1261

Offset: 1

R. J. Mathar, Sep 11 2006

nonn

The first duplicate is (-23,71,588),(14,70,588), the second (-119,140,1029),(49,98,1029). A033430(m) and A000578(k) are subsets since (x,y,z)=(2m,2m,4m^3) or (x,y,z)=(0,k^2,k^3) solve x^3+y^3=z^2. The "leakage" problem of A103254 can be avoided by introducing s=x+y and d=y-x and searching for solutions of the transformed equation s(s^2+3d^2)=4z^2 over all positive divisors s of 4z^2.

			(x,y,z)=(0,1,1),(1,2,3),(2,2,4),(0,4,8),(-7,8,13),(4,8,24),(0,9,27),(-6,10,28),
(8,8,32),(-7,14,49),(0,16,64),(9,18,81),(7,21,98),(-28,32,104).

Cf. A103254, A103255.

A217735 a(n) = the least positive integer m such that n^3 + m^3 is a square, or 0, if (presumably) there is no such m.

Original entry on oeis.org

2, 1, 0, 8, 0, 0, 21, 4, 18, 65, 37, 0, 0, 70, 0, 32, 0, 9, 0, 0, 7, 26, 1177, 0, 50, 22, 0, 84, 0, 0, 0, 16, 88, 450, 665, 72, 11, 1178, 0, 260, 0, 0, 0, 148, 0, 2, 0, 0, 98, 25, 0, 0, 0, 0, 0, 65, 112, 0, 0, 0, 0, 0, 189, 128, 10, 0, 0, 0, 0, 14, 0, 36, 0, 0, 0, 0, 0, 273, 0, 0, 162, 0, 0, 28, 0, 602, 0, 33, 0, 585, 65, 4708, 0, 0, 1121

Offset: 1

Zak Seidov, Mar 22 2013

nonn

All zero terms are suggestive(?). All positive terms are certain.

			n=1: 1^3+2^3=3^2
n=2: 2^3+1^3=3^2
n=4: 4^3+8^3=24^2
n=7: 7^3+21^3=98^2

Giovanni Resta, Table of n, a(n) for n = 1..1000 (see comments in b-file)

Cf. A103254 (values of n such that a(n) > 0).

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

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A103268 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^5.

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A121980 Positive integers z, without duplication, in x^3+y^3=z^2.

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A217735 a(n) = the least positive integer m such that n^3 + m^3 is a square, or 0, if (presumably) there is no such m.

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