A081119 -id:A081119 - cp's OEIS frontend

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).

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

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

Original entry on oeis.org

0, 9, 225, 14400, 278784, 16769025, 322382025, 19356600384, 372051201600, 22337675375625, 429347532814209, 25777663981977600, 495466706924481600, 29747402099825117409, 571768151330225342025, 34328476252406392070400, 659819951198501829398784, 39615031848108328736769225, 761431651915943270106720225, 45715712424248689455481003584, 878691466491082103705616000000

Offset: 0

Vladimir Pletser, Apr 30 2017

Numbers a(n) which are the square of triangular number T(b(n)), 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 parameters K of the 3rd degree Diophantine Bachet-Mordell equation y^2=x^3+K, with x= T(b(n)) = A006454(n) and y= T(b(n))* sqrt(T(b(n))+1) = A285955(n).
Also: a(n) = (A000217(A006451(n)))^2 or a(n) = A006454(n)^2.

			For n=2, b(n)=5, a(n)=225.
For n=5, b(n)=90, a(n)= 16769025.
For n = 3, A006451(n) = 15. Therefore, A000217(A006451(n)) = A000217(15) = 120 and (A000217(A006451(n)))^2 = (A000217(15))^2 = (120)^2 = 14400.

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..500
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. A285955, A006454, A000217, A006451, A081119, A054504.

restart: bm2:=-1: bm1:=0: bp1:=2: bp2:=5: print ('0,0','1,9','2,225'); for n from 3 to 1000 do b:= 8*sqrt((bp1^2+bp1)/2+1)+bm2; a:=(b*(b+1)/2)^2; 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)) (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)^2.
Empirical g.f.: 9*x*(1 + 24*x + 387*x^2 + 864*x^3 + 387*x^4 + 24*x^5 + x^6) / ((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 + 6*x + x^2)*(1 + 34*x + x^2)). - Colin Barker, Apr 30 2017

A356699 Numbers k such that Mordell's equation y^2 = x^3 + k has a record number of integral solutions.

Original entry on oeis.org

1, 8, 9, 17, 225, 1025

Offset: 1

Jianing Song, Aug 23 2022

Indices of records in A081119. For the record values see A356701.

			1025 is a term since the equation y^2 = x^3 + 1025 has 32 integral solutions, and the number of solutions to y^2 = x^3 + k is less than 32 for 0 < k < 1025.

Cf. A081119, A081120, A356700, A356701, A356702.

A356700 Numbers k such that Mordell's equation y^2 = x^3 - k has a record number of integral solutions.

Original entry on oeis.org

1, 2, 4, 28, 116, 207, 431, 2351, 3807

Offset: 1

Jianing Song, Aug 23 2022

Indices of records in A081120. For the record values see A356702.

			207 is a term since the equation y^2 = x^3 + 207 has 14 integral solutions, and the number of solutions to y^2 = x^3 - k is less than 14 for 0 < k < 207.

Cf. A081119, A081120, A356699, A356701, A356702.

A356702 Records values in A081120.

Original entry on oeis.org

1, 2, 4, 6, 8, 14, 18, 20, 22

Offset: 1

Jianing Song, Aug 23 2022

			a(9) = 22 since A356700(9) = 3807, and the equation y^2 = x^3 - 3807 has 22 integral solutions.

Cf. A081119, A081120, A356699, A356700, A356701.

A318932 Numbers k such that the Diophantine equation y^2 - k = x^3 has a solution.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 12, 15, 16, 17, 18, 19, 22, 24, 25, 26, 27, 28, 30, 31, 33, 35, 36, 37, 38, 40, 41, 43, 44, 48, 49, 50, 52, 54, 55, 56, 57, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 82, 89, 91, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117

Offset: 1

N. J. A. Sloane, Sep 12 2018

nonn

Hemer gives all values of k <= 100.

Numbers k such that Mordell's equation y^2 = x^3 + k has integral solutions. - Giorgos Kalogeropoulos, Mar 04 2021

Hugo Pfoertner, Table of n, a(n) for n = 1..1000
Ove Hemer, Notes on the diophantine equation y^2 - k = x^3, Arkiv för Matematik 3.1 (1954): 67-77.

Complement of A054504.

Cf. A081119, A318933.

More terms from Giorgos Kalogeropoulos, Mar 04 2021

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]

A329922 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives y-values).

Original entry on oeis.org

1, 1, 2, 2, 2, 0, 0, 3, 3, 3, 0, 2, 0, 0, 4, 4, 4, 19, 12, 0, 0, 7, 0, 5, 5, 5, 0, 6, 0, 83, 2, 0, 5, 0, 6, 6, 6, 37, 0, 16, 7, 0, 4, 6, 0, 0, 0, 7, 7, 7, 0, 5, 0, 9, 28, 8, 7, 0, 0, 0, 0, 0, 8, 8, 8, 0, 0, 2, 0, 0, 14, 8, 9, 0, 0, 7, 0, 0, 302, 9, 9, 9, 0, 0, 0, 0, 0, 0, 9, 0, 8, 10, 0, 11, 0, 0, 77, 21, 10, 10, 10, 0, 0, 0, 13, 59, 48, 10, 0, 0, 0, 29, 11, 0, 0, 0, 12, 0, 386, 11

Offset: 1

Jean-François Alcover, Nov 24 2019

nonn

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) = 19.

See A081119.

Jean-François Alcover, Table of n, a(n) for n = 1..10000

Cf. A054504, A081119 (number of solutions), A329921 (x-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][[2]];
a /@ Range[120]

A364248 For n >= 3, a(n) is the least r >= 0 such that the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2 has an integer solution.

Original entry on oeis.org

0, 2, 5, 9, 14, 0, 27, 5, 44, 12, 65, 21, 0, 32, 119, 9, 152, 35, 21, 77, 230, 0, 275, 117, 54, 14, 377, 41, 434, 32, 55, 221, 0, 27, 629, 285, 52, 20, 779, 49, 860, 11, 21, 437, 1034, 0, 1127, 75, 34, 182, 1325, 27, 110, 159, 19, 725, 1652, 10, 1769, 837, 0, 320, 195, 99, 2144, 374

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 n >= 4, n*(n - 3)/2 >= a(n) >= 2.

			n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47. Least r is 9 thus a(6) = 9 and [y, n, n+r] is [21, 6, 15].
n = 7: y^2 = 7^3 + (7 + r)^2 is valid for r = 14, 164. Least r is 14 thus a(7) = 14 and [y, n, n+r] is [28, 7, 21].

Robin Visser, Table of n, a(n) for n = 3..10000
Josef Gebel, Attila Pethő, and Horst G. Zimmer, Computing S-integral points on elliptic curves, Rend. Circ. Matem. Palermo 30 (1910), 377-407. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg.
The L-functions and modular forms database, LMFDB
Joseph H. Silverman, The Arithmetic of Elliptic Curves
Eric Weisstein's World of Mathematics, Mordell Curve

Cf. A000040, A000096, A000217, A002061, A005563, A006254, A054504, A081119, A364421.

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

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

a(p) = p*(p - 3)/2, p >= 5 a prime number.

a(n) = 0 for n from A005563.

cp's OEIS Frontend

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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Formula

Extensions

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

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A356699 Numbers k such that Mordell's equation y^2 = x^3 + k has a record number of integral solutions.

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A356700 Numbers k such that Mordell's equation y^2 = x^3 - k has a record number of integral solutions.

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A356702 Records values in A081120.

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A318932 Numbers k such that the Diophantine equation y^2 - k = x^3 has a solution.

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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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Mathematica

A329922 Integral solutions to Mordell's equation y^2 = x^3 - n with minimal absolute value of x (a(n) gives y-values).

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Mathematica

A364248 For n >= 3, a(n) is the least r >= 0 such that the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2 has an integer solution.

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A364248 For n >= 3, a(n) is the least r >= 0 such that the elliptic equation y^2 = n^3 + n^2 + 2rn + r^2 has an integer solution.