A134109 -id:A134109 - cp's OEIS frontend

A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.

1, 2, 0, 4, 0, 0, 4, 1, 0, 0, 4, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 1, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 2, 0, 6, 4, 2, 0, 0, 0, 4, 2, 4, 2, 0, 0, 0, 4, 2, 0, 4, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 4, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 6

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Pethö, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A081121 gives n for which there are no integral solutions. See A081119 for the number of integral solutions to y^2 = x^3 + n.
From Jianing Song, Aug 24 2022: (Start)
If A060951(n) = 0 (namely the elliptic curve y^2 = x^3 - n has rank 0), then:
- a(n) = 2 if n is of the form 432*t^6;
- a(n) = 1 if n is a cube;
- 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 A060951(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)

			a(4)=4 refers to (x,y) = (2,+-2) and (5,+-11).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

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 based on the work of J. Gebel.]
M. A. Bennett and A. Ghadermarzi, Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica. 110:3 (1998): 335-367.
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]
Joseph H. Silverman, The Arithmetic of Elliptic Curves.
Eric Weisstein's World of Mathematics, Mordell Curve.
Wikipedia, Mordell curve.

Cf. A081119, A081121. See A134109 for another version.

(* This naive approach gives correct results up to n=1000 *) xmax[] = 10^4; Do[ xmax[n] = 10^5, {n, {366, 775, 999}}]; Do[ xmax[n] = 10^6, {n, {207, 307, 847}}]; f[n] := (x = Floor[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, Mar 06 2012 *)

Edited by Max Alekseyev, Feb 06 2021

A134108 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.

Original entry on oeis.org

3, 1, 1, 1, 1, 0, 0, 4, 5, 1, 0, 2, 0, 0, 2, 1, 8, 1, 1, 0, 0, 1, 0, 4, 1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 1, 4, 3, 1, 0, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 3, 0, 0, 0, 0, 0, 2, 3, 4, 0, 0, 2, 0, 0, 1, 1, 6, 0, 0, 1, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 4, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 6, 2, 0, 0, 0, 1

Offset: 1

Klaus Brockhaus, Oct 08 2007, Oct 14 2007

nonn

a(n) = A081119(n)/2 if A081119(n) is even, (A081119(n)+1)/2 if A081119(n) is odd (i.e. if n is a cubic number).

Comment from T. D. Noe, Oct 12 2007: In sequences A134108 (this entry) and A134109 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 + 1 has solutions (y, x) = (0, -1), (1, 0) and (3, 2), hence a(1) = 3.
y^2 = x^3 + 6 has no solutions, hence a(6) = 0.
y^2 = x^3 + 17 has 8 solutions (see A029727, A029728), hence a(17) = 8.
y^2 = x^3 + 27 has solution (y, x) = (0, -3), hence a(27) = 1.

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. A081119, A029727, A029728, A134109.

[ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, n])) }: n in [1..122] ];

(* 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_] := a[n] = (fn = f[n]; an = If[fn == {}, 0, 2 Length[fn] - If[First[fn] == 0, 1, 0]]; If[EvenQ[an], an/2, (an + 1)/2]); Table[ Print["a[", n, "] = ", a[n] ]; a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 20 2012 *)
A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {, }][[All, 2]];
a[n_] := With[{an = A081119[[n]]}, If[EvenQ[an], an/2, (an + 1)/2]];
a /@ Range[10000] (* Jean-François Alcover, Nov 24 2019 *)

A329921 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives x-values).

Original entry on oeis.org

0, -1, 1, 0, -1, 0, 0, 1, 0, -1, 0, -2, 0, 0, 1, 0, -1, 7, 5, 0, 0, 3, 0, 1, 0, -1, -3, 2, 0, 19, -3, 0, -2, 0, 1, 0, -1, 11, 0, 6, 2, 0, -3, -2, 0, 0, 0, 1, 0, -1, 0, -3, 0, 3, 9, 2, -2, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -4, 0, 0, 5, -2, 2, 0, 0, -3, 0, 0, 45, 1, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, -3, 2, 0, 3

Offset: 1

Jean-François Alcover, Nov 24 2019

sign

Conventionally, no solution is indicated by (x,y) = (0,0).

			For n=12, the "min |x|" solution is 2^2 = (-2)^3+12, hence xy(12) = [-2,2] and a(12) = -2;
for n=18, it is 19^2  = 7^3 + 18, hence xy(18) = [7,19] and a(18) = 7.

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

Cf. A054504, A081119 (number of solutions), A134109, A329922 (y-values).

A081119 = Cases[Import["https://oeis.org/A081119/b081119.txt", "Table"], {, }][[All, 2]];
r[n_, x_] := Reduce[y >= 0 && y^2 == x^3 + n, y, Integers];
xy[n_] := If[A081119[[n]] == 0, {0, 0}, For[x = 0, True, x++, rn = r[n, x]; If[rn =!= False, Return[{x, y} /. ToRules[rn]]; Break[]]; rn = r[n, -x]; If[rn =!= False, Return[{-x, y} /. ToRules[rn]]; Break[]]]];
a[n_] := xy[n][[1]];
a /@ Range[120]

A228105 a(n) = 432*n^6.

Original entry on oeis.org

0, 432, 27648, 314928, 1769472, 6750000, 20155392, 50824368, 113246208, 229582512, 432000000, 765314352, 1289945088, 2085181488, 3252759552, 4920750000, 7247757312, 10427429808, 14693280768, 20323820592, 27648000000, 37050964272, 48980118528

Offset: 0

Arkadiusz Wesolowski, Aug 10 2013

For any n > 0, the equation y^2 = x^3 - a(n) has exactly one solution in natural numbers (x = 12*n^2 and y = 36*n^3).

			a(2) = 432*2^6 = 27648.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A134109.

Magma
```
[432*n^6 : n in [0..22]];
```
Maple
```
seq(432*n^6, n=0..22);
```

Table[432*n^6, {n, 0, 22}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,432,27648,314928,1769472,6750000,20155392},40] (* Harvey P. Dale, Apr 06 2018 *)

concat(0, Vec(432*x*(1 + x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Dec 11 2017

a(n) = A008585(n)*A008591(n)*A016744(n).
G.f.: 432*x*(1 + x)*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. - Colin Barker, Dec 11 2017

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A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.

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A134108 Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 + n.

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A329921 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives x-values).

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A228105 a(n) = 432*n^6.

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