A029728 -id:A029728 - cp's OEIS frontend

A029727 Complete list of solutions to y^2 = x^3 + 17; sequence gives y values.

3, 4, 5, 9, 23, 282, 375, 378661

Offset: 1

L. J. Mordell, Diophantine Equations, Ac. Press, p. 246.

See A029728 for further comments and references.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0,17])) ]); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

r[x_] := Reduce[y > 0 && y^2 == x^3 + 17, y, Integers]; y /. ToRules /@ Select[Table[r[x], {x, -2, 10000}], # =!= False & ] (* Jean-François Alcover, Sep 07 2011 *)

[i[1] for i in EllipticCurve([0, 17]).integral_points()] # Seiichi Manyama, Aug 25 2019

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 *)

A134042 Complete list of solutions to y^2 = x^3 + 113; sequence gives x values.

Original entry on oeis.org

-4, 2, 8, 11, 26, 422

Offset: 1

Artur Jasinski, Oct 03 2007

For corresponding y values and examples see A134043.

Cf. A134043, A029728, A080761.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 113])) ]); /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 113]).integral_points()] # Seiichi Manyama, Aug 25 2019

Edited and corrected by Klaus Brockhaus, Oct 04 2007

A134105 Complete list of solutions to y^2 = x^3 + 297; sequence gives x values.

Original entry on oeis.org

-6, -2, 3, 4, 12, 34, 48, 1362, 93844

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134104.

The parameter -297 of the curve corresponds to A200218(1). a(9)=A200216(1). - Artur Jasinski, Nov 29 2011

Cf. A134104, A134103, A134107, A029728, A134042, A134074.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029728 */

sol[x_] := Solve[y > 0 && x^3 - y^2 == -297, y, Integers];
Reap[For[x = 1, x < 10^5, x++, sx = sol[x]; If[sx != {}, xy = {x, y} /. sx[[1]]; Print[xy]; Sow[xy]]; sx = sol[-x]; If[sx != {}, xy = {-x, y} /. sx[[1]]; Print[xy]; Sow[xy]]]][[2, 1]][[All, 1]] // Sort (* Jean-François Alcover, Feb 07 2020 *)

[i[0] for i in EllipticCurve([0, 297]).integral_points()] # Seiichi Manyama, Aug 26 2019

A134107 Complete list of solutions to y^2 = x^3 - 207; sequence gives x values.

Original entry on oeis.org

6, 12, 18, 31, 312, 331, 367806

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134106.

Cf. A134106, A134103, A134105, A029728, A134042, A134074.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, -207]).integral_points()] # Seiichi Manyama, Aug 25 2019

A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

Original entry on oeis.org

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151

Offset: 1

Cino Hilliard, Mar 10 2003

nonn

From Artur Jasinski, Oct 03 2007: (Start)
Some numbers have multiple partitions:
  8 = 4^2 - 8^3 = 312^2 - 46^3,
  9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)
This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

			8 is in the sequence since 3^2 = 1^3 + 8.

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. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k.

Complement of A080762.
Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.
Cf. A029727, A029728, A134042, A134043.

With[{nn=100},Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn],2],#>0&]],nn]] (* Harvey P. Dale, Jul 10 2012 *)

diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),print1(p" "); break) ) ) }

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007

A134074 Complete list of solutions to y^2 = x^3 + 73; sequence gives x values.

Original entry on oeis.org

-4, 2, 3, 6, 72, 356

Offset: 1

Klaus Brockhaus, Oct 07 2007

For corresponding y values and examples see A134073.

Cf. A134073, A029728, A134042.

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 73])) ]); /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 73]).integral_points()] # Seiichi Manyama, Aug 25 2019

A134103 Complete list of solutions to y^2 = x^3 + 225; sequence gives x values.

Original entry on oeis.org

-6, -5, 0, 4, 6, 10, 15, 30, 60, 180, 336, 351, 720114

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding y values and examples see A134102.

Cf. A134102, A134105, A134107, A029728, A134042, A134074.

{ x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029728 */

[i[0] for i in EllipticCurve([0, 225]).integral_points()] # Seiichi Manyama, Aug 26 2019

A268509 Numbers x such that x^3 = y^2 + z for some y and some nonzero z with -x < z < x.

Original entry on oeis.org

2, 3, 5, 13, 15, 17, 32, 35, 37, 40, 43, 46, 52, 56, 63, 65, 99, 101, 109, 136, 143, 145, 152, 158, 175, 190, 195, 197, 243, 255, 257, 312, 317, 323, 325, 331, 336, 351, 356, 366, 377, 399, 401, 422, 483, 485, 560, 568, 575, 577, 584, 592, 654, 675, 677, 717, 741, 783, 785, 799, 810, 891, 899, 901, 909, 937, 944, 978

Offset: 1

Daniel Mondot, Feb 06 2016

nonn

List of x such as x^3 is a near square (see examples).

Note that z = 17 appears often (see A029728).

			2^3 = 3^2 - 1;
3^3 = 5^2 + 2;
5^3 = 11^2 + 4;
13^3 = 47^2 - 12;
15^3 = 58^2 + 11;
17^3 = 70^2 + 13;
32^3 = 181^2 + 7;
35^3 = 207^2 + 26;
37^3 = 225^2 + 28;
40^3 = 253^2 - 9;
43^3 = 282^2 - 17;
46^3 = 312^2 - 8;
52^3 = 375^2 - 17;
56^3 = 419^2 + 55;
63^3 = 500^2 + 47;
65^3 = 524^2 + 49;
99^3 = 985^2 + 74.

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A029728, A253181, A268510.

#include 
#include 
#include 
#define MAX2 10000
/* list number x and y such that x^3 = y^2 ± delta (0 < delta < x) */
/* this generates A268509 and A268510 */
long long unsigned b,c,d;
long long signed ds;
unsigned long long list2[MAX2];
unsigned long long list3[MAX2];
long double b1, cd, dd;
void main(unsigned argc, char *argv[])
{
unsigned a, i;
  i=0;
  // I never actually calculate b^3 or c^2, but only b^(3/2) = c + ds
  // this allows me to indirectly check b^3 past 2^64
  for (b=0; b<100000000; ++b) // could go up to b<4294967295u; max
  {
    b1 = sqrtl(b);
    cd= b1 *(long double)b;
    c=(long long unsigned)(cd+(double)0.5);
    dd = 2 * c * (cd - c);
    if (dd<0) ds = (dd - 0.5);
    else ds = (dd + 0.5);
    d = llabs(ds);
    if (dA268509 */
  for (a=0; aA268510 */
  for (a=0; a

is(n)=my(t=abs(n^3-round(n^1.5)^2)); 0Charles R Greathouse IV, Feb 09 2016

A268510 Numbers x such that x^2 = y^3 + z (0 < abs(z) < y).

Original entry on oeis.org

3, 5, 11, 47, 58, 70, 181, 207, 225, 253, 282, 312, 375, 419, 500, 524, 985, 1015, 1138, 1586, 1710, 1746, 1874, 1986, 2315, 2619, 2723, 2765, 3788, 4072, 4120, 5511, 5644, 5805, 5859, 6022, 6159, 6576, 6717, 7002, 7320, 7970, 8030, 8669, 10615, 10681, 13252, 13537, 13788, 13860, 14113, 14404, 16725, 17537, 17615

Offset: 1

Daniel Mondot, Feb 06 2016

nonn

List of n such as n^2 is a near cube (see examples).
Numbers x such that x^2 = y^3 + 0 (e.g. 1000^2 = 100^3) are omitted.
Note that a delta of 17 appears often. See A029728.

			3^2 = 2^3 + 1
5^2 = 3^3 - 2
11^2 = 5^3 - 4
47^2 = 13^3 + 12
58^2 = 15^3 - 11
70^2 = 17^3 - 13
181^2 = 32^3 - 7
207^2 = 35^3 - 26
225^2 = 37^3 - 28
253^2 = 40^3 + 9
282^2 = 43^3 + 17
312^2 = 46^3 + 8
375^2 = 52^3 + 17
419^2 = 56^3 - 55
500^2 = 63^3 - 47
524^2 = 65^3 - 49
985^2 = 99^3 - 74

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A029728, A268509, A253181.

Select[Range@ 5000, Resolve@ Exists[{y, z}, And[Reduce[#^2 == (y^3 + z), {y, z}, Integers], 0 < Abs@ z < y]] &] (* Michael De Vlieger, Feb 07 2016 *)

cp's OEIS Frontend

A029727 Complete list of solutions to y^2 = x^3 + 17; sequence gives y values.

Views

Author

Keywords

References

Crossrefs

Programs

Magma

Mathematica

SageMath

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A134042 Complete list of solutions to y^2 = x^3 + 113; sequence gives x values.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

SageMath

Extensions

A134105 Complete list of solutions to y^2 = x^3 + 297; sequence gives x values.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

SageMath

A134107 Complete list of solutions to y^2 = x^3 - 207; sequence gives x values.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

SageMath

A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A134074 Complete list of solutions to y^2 = x^3 + 73; sequence gives x values.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

SageMath

A134103 Complete list of solutions to y^2 = x^3 + 225; sequence gives x values.

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma