A029727 -id:A029727 - cp's OEIS frontend

A029728 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.

-2, -1, 2, 4, 8, 43, 52, 5234

Offset: 1

Comments by Henri Cohen on the proof that the list of solutions is complete: (Start)
This is now completely standard. Cremona's mwrank program tells us that this is an elliptic curve of rank 2 with generators P1=(-2,3) and P2=(4,9).
We now apply the algorithm (essentially due to Tzanakis and de Weger) described in Nigel Smart's book on the algorithmic solution of Diophantine equations: using Sinnou David's bounds on linear forms in elliptic logarithms one finds that if P is an integral point then P=aP1+bP2 for |a| and |b| less than a huge bound B (typically 10^100, sometimes more, I didn't do the computation here), but the main point is that B is completely explicit. One then uses the LLL algorithm: this is crucial.
A first application reduces the bound to 200, say, then a second application to 20 and sometimes a third to 12 (at this point it is not necessary). Then a very small search gives all the possible integer points. (End)

L. J. Mordell, Diophantine Equations, Ac. Press, p. 246.
T. Nagell, Einige Gleichungen von der Form ay^2+by+c=dx^3, Vid. Akad. Skrifter Oslo, Nr. 7 (1930).
Silverman, Joseph H. and John Tate, Rational Points on Elliptic Curves. New York: Springer-Verlag, 1992.

Cf. A029727 (y values).
x values of solutions to y^2 = x^3 + a*x + b;
A134107 (a= 0, b=-207),
A134074 (a= 0, b=  73),
A134042 (a= 0, b= 113),
A134103 (a= 0, b= 225),
A134105 (a= 0, b= 297),
A134167 (a= 0, b=1025),
A316456 (a=-7, b=  10),
A309071 (a=20, b=   0).

Sort([ p[1] : p in IntegralPoints(EllipticCurve([0,17])) ]); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

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

[i[0] 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 *)

A134043 Complete list of solutions to y^2 = x^3 + 113; sequence gives y values.

Original entry on oeis.org

7, 11, 25, 38, 133, 8669

Offset: 1

Artur Jasinski, Oct 03 2007

For corresponding x values see A134043.

			a(1)^2 = 7^2 = 49 = A134042(1)^3 + 113 = -64 + 113.
a(2)^2 = 11^2 = 121 = A134042(2)^3 + 113 = 8 + 113.
a(3)^2 = 25^2 = 625 = A134042(3)^3 + 113 = 512 + 113.
a(4)^2 = 38^2 = 1444 = A134042(4)^3 + 113 = 1331+ 113.
a(5)^2 = 133^2 = 17689 = A134042(5)^3 + 113 = 17576 + 113.
a(6)^2 = 8669^2 = 75151561 = A134042(6)^3 + 113 = 75151448 + 113.

Cf. A134042, A029727, A080761.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 113])) ]); /* adapted from A029727 */

(Program does not produce first two terms) a = {}; Do[k = n^2 - (Floor[n^(2/3)])^3; If[(k > 112) && (k < 113), AppendTo[a, n]], {n, 1, 100000}]; a

Edited and corrected by Klaus Brockhaus, Oct 04 2007

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

A134073 Complete list of solutions to y^2 = x^3 + 73; sequence gives y values.

Original entry on oeis.org

3, 9, 10, 17, 611, 6717

Offset: 1

Klaus Brockhaus, Oct 07 2007

For corresponding x values see A134074.

			a(1)^2 = 3^2 = 9 = A134074(1)^3 + 73 = -64 + 73.
a(2)^2 = 9^2 = 81 = A134074(2)^3 + 73 = 8 + 73.
a(3)^2 = 10^2 = 100 = A134074(3)^3 + 73 = 27 + 73.
a(4)^2 = 17^2 = 289 = A134074(4)^3 + 73 = 216+ 73.
a(5)^2 = 611^2 = 373321 = A134074(5)^3 + 73 = 373248+ 73.
a(6)^2 = 6717^2 = 45118089 = A134074(6)^3 + 73 = 45118016+ 73.

Cf. A134074, A029727, A134043.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 73])) ]); /* adapted from A029727 */

A134106 Complete list of solutions to y^2 = x^3 - 207; sequence gives y values.

Original entry on oeis.org

3, 39, 75, 172, 5511, 6022, 223063347

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding x values see A134107.

			a(1)^2 = 3^2 = 9 = A134107(1)^3 - 207 = 216 - 207.
a(2)^2 = 39^2 = 1521 = A134107(2)^3 - 207 = 1728 - 207.
a(3)^2 = 75^2 = 5625 = A134107(3)^3 - 207 = 5832 - 207.
a(4)^2 = 172^2 = 29584 = A134107(4)^3 - 207 = 29791 - 207.
a(5)^2 = 5511^2 = 30371121 = A134107(5)^3 - 207 = 30371328 - 207.
a(6)^2 = 6022^2 = 36264484 = A134107(6)^3 - 207 = 36264691 - 207.
a(7)^2 = 223063347^2 = 49757256774842409 = A134107(7)^3 - 207 = 49757256774842616 - 207.

Cf. A134107, A134102, A134104, A029727, A134043, A134073.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -207])) ]); /* adapted from A029727 */

[x[1] for x in EllipticCurve([0,-207]).integral_points()] # Charles R Greathouse IV, Aug 09 2024

A134102 Complete list of solutions to y^2 = x^3 + 225; sequence gives y values.

Original entry on oeis.org

3, 10, 15, 17, 21, 35, 60, 165, 465, 2415, 6159, 6576, 611085363

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding x values see A134103.

			a(1)^2 = 3^2 = 9 = A134103(1)^3 + 225 = -216 + 225.
a(2)^2 = 10^2 = 100 = A134103(2)^3 + 225 = -125 + 225.
a(3)^2 = 15^2 = 225 = A134103(3)^3 + 225 = 0 + 225.
a(4)^2 = 17^2 = 289 = A134103(4)^3 + 225 = 64 + 225.
a(5)^2 = 21^2 = 441 = A134103(5)^3 + 225 = 216 + 225.
a(6)^2 = 35^2 = 1225 = A134103(6)^3 + 225 = 1000 + 225.
a(7)^2 = 60^2 = 3600 = A134103(7)^3 + 225 = 3375 + 225.
a(8)^2 = 165^2 = 27225 = A134103(8)^3 + 225 = 27000 + 225.
a(9)^2 = 465^2 = 216225 = A134103(9)^3 + 225 = 216000 + 225.
a(10)^2 = 2415^2 = 5832225 = A134103(10)^3 + 225 = 5832000 + 225.
a(11)^2 = 6159^2 = 37933281 = A134103(11)^3 + 225 = 37933056 + 225.
a(12)^2 = 6576^2 = 43243776 = A134103(12)^3 + 225 = 43243551 + 225.
a(13)^2 = 611085363^2 = 373425320872841769 = A134103(13)^3 + 225 = 373425320872841544 + 225.

Cf. A134103, A134104, A134106, A029727, A134043, A134073.

{ x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 225])) ]) }; /* adapted from A029727 */

Select[Table[Sqrt[x^3+225],{x,-6,721000}],IntegerQ] (* Harvey P. Dale, Dec 25 2022 *)

A134104 Complete list of solutions to y^2 = x^3 + 297; sequence gives y values.

Original entry on oeis.org

9, 17, 18, 19, 45, 199, 333, 50265, 28748141

Offset: 1

Klaus Brockhaus, Oct 08 2007

For corresponding x values see A134105.

			a(1)^2 = 9^2 = 81 = A134105(1)^3 + 297 = -216 + 297.
a(2)^2 = 17^2 = 289 = A134105(2)^3 + 297 = -8 + 297.
a(3)^2 = 18^2 = 324 = A134105(3)^3 + 297 = 27 + 297.
a(4)^2 = 19^2 = 361 = A134105(4)^3 + 297 = 64 + 297.
a(5)^2 = 45^2 = 2025 = A134105(5)^3 + 297 = 1728 + 297.
a(6)^2 = 199^2 = 39601 = A134105(6)^3 + 297 = 39304 + 297.
a(7)^2 = 333^2 = 110889 = A134105(7)^3 + 297 = 110592 + 297.
a(8)^2 = 50265^2 = 2526570225 = A134105(8)^3 + 297 = 2526569928 + 297.
a(9)^2 = 28748141^2 = 826455610955881 = A134105(9)^3 + 297 = 826455610955584 + 297.

Cf. A134105, A134102, A134106, A029727, A134043, A134073.

Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 297])) ]); /* adapted from A029727 */

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, 2]] // Sort (* Jean-François Alcover, Feb 07 2020 *)

A124451 Complete list of solutions to y^2 = x^3 + 17; sequence gives y values.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

The solutions here are listed in the order given by Mordell. See A029728 and A029727 for a better version (with comments and references).

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

Cf. A124439 (x values). See A029728 for further comments and references.

A134166 Complete list of solutions to y^2 = x^3 + 1025; sequence gives y values.

Original entry on oeis.org

5, 30, 31, 32, 33, 45, 95, 255, 355, 513, 1930, 2139, 9419, 27905, 218796, 227805

Offset: 1

Klaus Brockhaus, Oct 11 2007

For corresponding x values see A134167.

			a(1)^2 = 5^2 = 25 = A134167(1)^3 + 1025 = -1000 + 1025.
a(2)^2 = 30^2 = 900 = A134167(2)^3 + 1025 = -125 + 1025.
a(3)^2 = 31^2 = 961 = A134167(3)^3 + 1025 = -64 + 1025.
a(4)^2 = 32^2 = 1024 = A134167(4)^3 + 1025 = -1 + 1025.
a(5)^2 = 33^2 = 1089 = A134167(5)^3 + 1025 = 64 + 1025.
a(6)^2 = 45^2 = 2025 = A134167(6)^3 + 1025 = 1000 + 1025.
a(7)^2 = 95^2 = 9025 = A134167(7)^3 + 1025 = 8000 + 1025.
a(8)^2 = 255^2 = 65025 = A134167(8)^3 + 1025 = 64000 + 1025.
a(9)^2 = 355^2 = 126025 = A134167(9)^3 + 1025 = 125000 + 1025.
a(10)^2 = 513^2 = 263169 = A134167(10)^3 + 1025 = 262144 + 1025.
a(11)^2 = 1930^2 = 3724900 = A134167(11)^3 + 1025 = 3723875 + 1025.
a(12)^2 = 2139^2 = 4575321 = A134167(12)^3 + 1025 = 4574296 + 1025.
a(13)^2 = 9419^2 = 88717561 = A134167(13)^3 + 1025 = 88716536 + 1025.
a(14)^2 = 27905^2 = 778689025 = A134167(14)^3 + 1025 = 778688000 + 1025.
a(15)^2 = 218796^2 = 47871689616 = A134167(15)^3 + 1025 = 47871688591 + 1025.
a(16)^2 = 227805^2 = 51895118025 = A134167(16)^3 + 1025 = 51895117000 + 1025.

Cf. A134167, A029727, A134043, A134073, A134102, A134104, A134106.

{ x : x in Sort([ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, 1025])) ]) }; /* adapted from A029727 */

Select[Table[Sqrt[1025+n^3],{n,-10,20000}],IntegerQ] (* Harvey P. Dale, Jan 21 2023 *)

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A029728 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.

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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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A134043 Complete list of solutions to y^2 = x^3 + 113; sequence gives y values.

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

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A134073 Complete list of solutions to y^2 = x^3 + 73; sequence gives y values.

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A134106 Complete list of solutions to y^2 = x^3 - 207; sequence gives y values.

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A134102 Complete list of solutions to y^2 = x^3 + 225; sequence gives y values.

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A134104 Complete list of solutions to y^2 = x^3 + 297; sequence gives y values.

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