A134108 -id:A134108 - cp's OEIS frontend

A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

5, 2, 2, 2, 2, 0, 0, 7, 10, 2, 0, 4, 0, 0, 4, 2, 16, 2, 2, 0, 0, 2, 0, 8, 2, 2, 1, 4, 0, 2, 2, 0, 2, 0, 2, 8, 6, 2, 0, 2, 2, 0, 2, 4, 0, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 2, 6, 0, 0, 0, 0, 0, 4, 5, 8, 0, 0, 4, 0, 0, 2, 2, 12, 0, 0, 2, 0, 0, 2, 8, 2, 2, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 0, 2, 0, 0, 2, 2, 2, 12

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A054504 gives n for which there are no integral solutions. See A081120 for the number of integral solutions to y^2 = x^3 - n.
a(n) is odd iff n is a cube. - Bernard Schott, Nov 23 2019
From Jianing Song, Aug 24 2022: (Start)
a(n) = 5 if n is a sixth power. Further more, if A060950(n) = 0 (namely the elliptic curve y^2 = x^3 + n has rank 0), then:
- a(n) = 2 if n is a square but not a sixth power;
- a(n) = 1 if n is a cube but not a sixth power;
- a(n) = 0 otherwise.
This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
- In all the other cases, the torsion group is trivial.
So a torsion point on y^2 = x^3 + n other than O is an integral point. If y^2 = x^3 + n has rank 0, then all the integral points on y^2 = x^3 + n are exactly the torsion points other than O.
Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060950(n) = 0: the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, so it has the same Mordell-Weil group (hence the same rank and isomorphic torsion group) as y^2 = x^3 + n. (End)

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (3) (1998), 335-367.

Jean-François Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and was based on the work of J. Gebel.]
Michael A. Bennett, Amir Ghadermarzi Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
Joseph H. Silverman, The Arithmetic of Elliptic Curves.
Eric Weisstein's World of Mathematics, Mordell Curve.
Wikipedia, Mordell curve.

Cf. A054504, A081120. See A134108 for another version.

(* This naive approach gives correct results up to n = 1000 *) xmax[] = 10^4; Do[xmax[n] = 10^5, {n, {297, 377, 427, 885, 899}}]; Do[xmax[n] = 10^6, {n, {225, 353, 618 }}]; f[n] := (x = -Ceiling[n^(1/3)]-1; s = {}; While[x <= xmax[n], x++; y2 = x^3 + n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n];  If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0] ]); Table[an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Oct 18 2011 *)

Edited by Max Alekseyev, Feb 06 2021

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

A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 3, 2, 1, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 0, 3

Offset: 1

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

nonn

a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

			y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.
y^2 = x^3 - 5 has no solutions, hence a(5) = 0.
y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.
y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.

Jean-François Alcover, Table of n, a(n) for n = 1..10000
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]
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A081120, A134106, A134107, A134108.

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];

A081120 = Cases[Import["https://oeis.org/A081120/b081120.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081120[[n]]}, If[EvenQ[an], an/2, (an+1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 28 2019 *)

A134223 Values n for which integer solutions of Mordell curve y^2=x^3+n have 4 different values of x (or equivalently of nonnegative y).

Original entry on oeis.org

8, 24, 36, 65, 80, 89, 108, 145, 161, 233, 260, 353, 377, 441, 449, 505, 521, 528, 537, 649, 681, 737, 745, 784, 792, 1100, 1116, 1224, 1296, 1412, 1513, 1536, 1548, 1585, 1753, 1897, 1961, 2025, 2033, 2185, 2250, 2305, 2404, 2521, 2537, 2745, 2793, 2852

Offset: 1

Artur Jasinski, Oct 14 2007

nonn

Union of A179151 and A179152.

Cf. A134108, A134220, A134221, A134222.

Numbers n such that A134108(n) = Floor((A081119(n)+1)/2) = 4.

Edited and extended by Ray Chandler, Jul 12 2010

A134220 Values n for which integer solutions of Mordell curve y^2=x^3+n have a single value of x (or equivalently of nonnegative y).

Original entry on oeis.org

2, 3, 4, 5, 10, 16, 18, 19, 22, 25, 26, 27, 30, 31, 33, 35, 38, 40, 41, 43, 48, 49, 50, 52, 54, 55, 56, 71, 72, 76, 79, 81, 82, 91, 92, 94, 97, 98, 99, 105, 106, 107, 112, 117, 119, 120, 122, 125, 126, 127, 131, 132, 134, 136, 138, 142, 143, 144, 150, 151, 152, 154, 156

Offset: 1

Artur Jasinski, Oct 14 2007, Oct 16 2007

nonn

Union of A179145 and A179146.

Cf. A134108, A134221, A134222, A134223.

Numbers n such that A134108(n) = Floor((A081119(n)+1)/2) = 1.

Edited and extended by Ray Chandler, Jul 12 2010

A134221 Values n for which integer solutions of Mordell curve y^2=x^3+n have 2 different values of x (or equivalently of nonnegative y).

Original entry on oeis.org

12, 15, 28, 44, 63, 68, 101, 121, 128, 148, 168, 197, 198, 204, 208, 220, 232, 248, 269, 294, 337, 343, 346, 350, 369, 404, 409, 443, 481, 485, 492, 540, 556, 561, 575, 618, 640, 656, 659, 701, 702, 716, 740, 757, 768, 775, 785, 804, 829, 850, 857, 868, 885

Offset: 1

Artur Jasinski, Oct 14 2007

nonn

Union of A179147 and A179148.

Cf. A134108, A134220, A134222, A134223.

Numbers n such that A134108(n) = Floor((A081119(n)+1)/2) = 2.

Edited and extended by Ray Chandler, Jul 12 2010

A134222 Values n for which integer solutions of Mordell curve y^2=x^3+n have 3 different values of x (or equivalently of nonnegative y).

Original entry on oeis.org

1, 37, 57, 64, 129, 141, 164, 169, 171, 196, 281, 289, 359, 392, 414, 427, 433, 464, 513, 516, 577, 593, 612, 625, 633, 665, 684, 721, 729, 730, 793, 801, 841, 849, 899, 940, 953, 964, 1000, 1001, 1081, 1090, 1153, 1169, 1233, 1252, 1289, 1297, 1380, 1441

Offset: 1

Artur Jasinski, Oct 14 2007

nonn

Union of A179149 and A179150.

Cf. A134108, A134220, A134221, A134223.

Numbers n such that A134108(n) = Floor((A081119(n)+1)/2) = 3.

Edited and extended by Ray Chandler, Jul 12 2010

A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 3, 4, 2, 8, 2, 4, 8, 5, 2, 7, 2, 9, 8, 4, 2, 15, 3, 4, 5, 10, 2, 15, 2, 7, 8, 4, 8, 17, 2, 4, 8, 15, 2, 15, 2, 10, 14, 4, 2, 22, 3, 7, 8, 9, 2, 10, 8, 15, 8, 4, 2, 38, 2, 4, 14, 8, 8, 15, 2, 9, 7, 16, 2, 27, 2, 4, 13, 9, 8, 15, 2, 22, 6, 4, 2, 39, 8, 4, 7, 16, 2, 27, 8, 10

Offset: 3

Ctibor O. Zizka, Sep 01 2023

nonn

The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section.
For a prime number p >= 5, the equation y^2 = p^3 + (p + r)^2 has 2 solutions, r_1 = p*(p - 3)/2 and r_2 = (p + 1)*(p^2 - p - 1)/2.
Factoring the equation y^2 = n^3 + n^2 + 2*r*n + r^2 yields (y+n+r)*(y-n-r) = n^3, which implies y+n+r = d and y-n-r = n^3/d for some divisor d of n^3.  Thus a(n) is the number of divisors d of n^3 such that (d-n^3/d)/2 - n is a nonnegative integer.  This resolves some of Thomas Scheuerle's conjectures. - Robin Visser, Sep 30 2023

			n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47, thus a(6) = 3. The 3 solutions [y, n, n+r] are [21, 6, 15], [29, 6, 25], [55, 6, 53].

Robin Visser, Table of n, a(n) for n = 3..10000
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A081119, A134108, A190300, A196226, A245685, A364248.

a(n) = length(select((x) -> x[1] >= 0 && x[2] >= n, thue(thueinit(x^2-1,1),n^3),1)) \\ Thomas Scheuerle, Sep 03 2023

def a(n):
    num_sols = 0
    for d in Integer(n^3).divisors():
        if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): num_sols += 1
    return num_sols  # Robin Visser, Sep 30 2023

a(p) = 2 for p prime >= 5, see Comments.
From Thomas Scheuerle, Sep 04 2023: (Start)
Conjecture: a(A190300(n)) = 3.
Conjecture: a(A196226(n)) = 4.
Conjecture: a(p^3) = 5 if p is an odd prime.
Conjecture: a(2*p^2) = 7 if p is an odd prime. But there exist other cases too, for example a(3*23) = 7.
Conjecture: a(prime(n)^prime(n)) = A245685(n - 1) - 1. (End)

a(61)-a(92) from Thomas Scheuerle, Sep 01 2023

cp's OEIS Frontend

A081119 Number of integral solutions to Mordell's equation y^2 = x^3 + n.

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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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A134109 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

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A134223 Values n for which integer solutions of Mordell curve y^2=x^3+n have 4 different values of x (or equivalently of nonnegative y).

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A134220 Values n for which integer solutions of Mordell curve y^2=x^3+n have a single value of x (or equivalently of nonnegative y).

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A134221 Values n for which integer solutions of Mordell curve y^2=x^3+n have 2 different values of x (or equivalently of nonnegative y).

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A134222 Values n for which integer solutions of Mordell curve y^2=x^3+n have 3 different values of x (or equivalently of nonnegative y).

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A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2.

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A364421 For n >= 3, r >= 0, y an integer, a(n) is the number of integral solutions to the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2.