A081119 -id:A081119 - cp's OEIS frontend

A031508 a(n) = smallest k > 0 such that the elliptic curve y^2 = x^3 - k has rank n, or -1 if no such k exists.

1, 2, 11, 174, 2351, 28279, 975379

Offset: 0

See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n. - Jonathan Sondow, Sep 06 2013
See A060951 for the rank of y^2 = x^3 - n. - Jonathan Sondow, Sep 10 2013
Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - Charles R Greathouse IV, Sep 10 2013
a(7) <= 56877643. a(8) <= 2520963512. a(9) <= 463066403167. a(10) <= 56736325657288. a(11) <= 46111487743732324. a(12) <= 6533891544658786928. See Table 3.3 in [Womack 2003]. - Jose Aranda, Jun 30 2024
The three questions for arbitrary k, positive k, and negative k are not very far from each other because the curves for k and -27k are related by a 3-isogeny and therefore have the same rank. It would be most natural to ask for the minimal |k| for k of either sign [see A373795].  - Noam D. Elkies,  Jul 02 2024
a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - N. J. A. Sloane, Jul 05 2024

			From _M. F. Hasler_, Jul 01 2024: (Start)
Sequence A060951 = (0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, ...) gives the analytic rank of the elliptic curve y^2 = x^3 - k for k = 1, 2, 3, ...
We can see that:
  - the smallest k that gives rank 0 is k = 1 = a(0);
  - the smallest k that gives rank 1 is k = 2 = a(1);
  - the smallest k that gives rank 2 is k = 11 = a(2); etc. (End)

Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

J. E. Cremona, Elliptic Curve Data
Noam D. Elkies and Zev Klagsbrun, New rank records for elliptic curves having rational torsion, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.
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]
J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064.
Tom Womack, Explicit Descent on Elliptic Curves, PhD thesis, University of Nottingham, July 2003
Tom Womack, Minimal-known positive and negative k for Mordell curves of given rank (personal web page, latest available snapshot on web.archive.org from Jan. 2017), last modified 10/2002.

Cf. A031507, A373795.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019

{A031508(n)=for(k=1,oo, ellrank(ellinit([0, -k]))[1]==n && return(k))} \\ M. F. Hasler, Jul 01 2024

a(n) = min { k >= 1 | A060951(k) == n }. - M. F. Hasler, Jul 01 2024

Definition clarified by Jonathan Sondow, Oct 26 2013.

Escape clause added to definition by N. J. A. Sloane, Jun 29 2024, because, as John Cremona reminds me, it is not known if k always exists.

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

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

A285955 Numbers a(n) = T(b(n))*sqrt(T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = y solutions of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and K = (T(b(n)))^2= A285985(n).

Original entry on oeis.org

0, 6, 60, 1320, 12144, 262080, 2405970, 51894744, 476378760, 10274921850, 94320640056, 2034382775040, 18675010652760, 402797515372356, 3697557790357470, 79751873665825680, 732097767490332144, 15790468188346521390, 144951660405354891060, 3126432949419110989944

Offset: 0

Vladimir Pletser, Apr 29 2017

Numbers a(n) which are the products of the triangular number T(b(n)) and the square root of this triangular number plus one, sqrt(T(b(n))+1), where  b(n) is the sequence A006451(n) of numbers n such that T(n)+1 is a square.
This sequence a(n) gives also the y solutions of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with x= T(b(n)) = A006454(n) and K = (T(b(n)))^2 = A285985(n), where T(b(n)) is the triangular number of b(n)= A006451(n).
Also: A000217(A006451(n)) * sqrt(A000217(A006451(n))+1).

			For n=2, b(n)=5, a(n)=60.
For n=5, b(n)=90, a(n)= 262080.
For n = 3, A006451(n) = 15. Therefore, A000217(A006451(n)) = A000217(15) = 120. This gives A000217(A006451(n)) * sqrt(A000217(A006451(n)) + 1) = 120 * sqrt(120 + 1) = 1320. - _David A. Corneth_, Apr 29 2017

V. Pletser, On some solutions of the Bachet-Mordell equation for large parameter values, to be submitted, April 2017.

Vladimir Pletser, Table of n, a(n) for n = 0..1000
M.A. Bennett and A. Ghadermarzi, Data on Mordell's curve.
Michael A. Bennett, Amir Ghadermarzi, Mordell's equation : a classical approach, arXiv:1311.7077 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A285985, A006454, A000217, A006451, A081119, A054504.

restart: bm2:=-1: bm1:=0: bp1:=2: bp2:=5: print (‘0,0’,’1,6’,’2,60’); for n from 3 to 1000 do b:= 8*sqrt((bp1^2+bp1)/2+1)+bm2; a:=(b*(b+1)/2)* sqrt((b*(b+1)/2)+1); print(n,a); bm2:=bm1; bm1:=bp1; bp1:=bp2; bp2:=b; end do:

Since b(n) = 8*sqrt(T(b(n-2))+1)+ b(n-4) = 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4), with b(-1)=-1, b(0)=0, b(1)=2, b(2)=5 (see A006451) and a(n) = T(b(n))*sqrt(T(b(n))+1) (this sequence), one has :
a(n) = ([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)* sqrt(([8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)]*[ 8*sqrt((b(n-2)*(b(n-2)+1)/2)+1)+ b(n-4)+1]/2)+1).
Empirical g.f.: 6*x*(1 - x)*(1 + 11*x + 27*x^2 + 11*x^3 + x^4) / ((1 + 14*x - x^2)*(1 + 2*x - x^2)*(1 - 2*x - x^2)*(1 - 14*x - x^2)). - Colin Barker, Apr 30 2017

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.

cp's OEIS Frontend

A031508 a(n) = smallest k > 0 such that the elliptic curve y^2 = x^3 - k has rank n, or -1 if no such k exists.

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

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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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A285955 Numbers a(n) = T(b(n))*sqrt(T(b(n))+1), where T(b(n)) is the triangular number of b(n)= A000217(b(n)) and b(n)=A006451(n). Also a(n) = y solutions of the Bachet Mordell equation y^2=x^3+K, where x= T(b(n)) = A006454(n) and K = (T(b(n)))^2= A285985(n).

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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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