A054504 -id:A054504 - cp's OEIS frontend

A177986 Numbers n such that quartic curve y^2=x^4+n have integral points.

Original entry on oeis.org

1, 3, 4, 8, 9, 15, 16, 19, 20, 24, 25, 28, 33, 35, 36, 40, 48, 49, 51, 63, 64, 65, 68, 73, 80, 81, 84, 99, 100, 104

Offset: 1

Artur Jasinski, May 16 2010

nonn

To this sequence belonging as subset all perfect squares and squares-1.

Complement to this sequence see A177987.

Cf. A054504, A081121.

A177987 Numbers n such that quartic equation y^2=x^4+n has no solution.

Original entry on oeis.org

2, 5, 6, 7, 10, 11, 12, 13, 14, 17, 18, 21, 22, 23, 26, 27, 29, 30, 31, 32, 34, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98

Offset: 1

Artur Jasinski, May 16 2010

nonn

Complement to this sequence see A177986.

			104 does not belong to this sequence because 27^2 = 5^4 + 104.

Cf. A054504, A081121.

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

A080762 Positive numbers not of the form y^2 - x^3, x and y >= 1.

Original entry on oeis.org

2, 4, 5, 6, 7, 10, 11, 13, 14, 16, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 34, 39, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 58, 59, 60, 61, 62, 66, 67, 69, 70, 72, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 93, 95, 96, 102, 103, 104, 109, 110, 111, 114, 115

Offset: 1

Cino Hilliard, Mar 10 2003

nonn

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 (positive or negative x and y). Hence, all of those numbers will be in this sequence. Additional terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

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, Integer points on Mordell curves
J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.

Complement of A080761.

diop(n,m) = {f=0; for(p=1,m, f=0; for(x=1,n, y=x*x*x+p; if(issquare(y),f=1); ); if(f==0,print1(p" ")) ) }

A125643 Squares and cubes (with repetition).

Original entry on oeis.org

0, 0, 1, 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681

Offset: 1

Zak Seidov, Oct 19 2006

nonn

Repeating terms are sixth powers: 0,1,64,729,... (A001014).

For numbers not appearing as a difference between a square and an adjacent cube in this list, see A054504 and A081121.

Zak Seidov, Table of n, a(n) for n = 1..1000.

Cf. A002760 (squares and cubes (without repetitions)).

Cf. A001014, A054504, A081121, A087285, A087286, A088017.

m=1681;cm=Floor[m^(1/3)];sm=Floor[Sqrt[m]];s=Range[0,sm]^2;c=Range[0,cm]^3;Sort[Join[s,c]] (* James C. McMahon, Dec 20 2024 *)

from math import isqrt
from sympy import integer_nthroot
def A125643(n):
    if n <= 4: return n-1>>1
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-2+x-integer_nthroot(x,3)[0]-isqrt(x)
    return bisection(f,n-2,n-2) # Chai Wah Wu, Oct 14 2024

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jul 14 2007

A177988 Numbers n such that quartic curve y^2=x^4-n has integral points.

Original entry on oeis.org

1, 7, 12, 15, 16, 17, 31, 32, 45, 49, 56, 60, 65, 71, 72, 77, 80, 81

Offset: 1

Artur Jasinski, May 16 2010

nonn

Complement to this sequence see A177989.

Numbers n such that quartic curve y^2=x^4+n has integral points. see A177986.

Cf. A054504, A081121, A177986, A177987, A177988, A177989.

Magma
```
IntegralQuarticPoints([1,0,0,0,-81]);
```

A177989 Numbers n such that quartic equation y^2=x^4-n has no integer solution.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 61, 62, 63, 64, 66, 67, 68, 69, 70, 73, 74, 75, 76, 78, 79

Offset: 1

Artur Jasinski, May 16 2010

nonn

Complement to this sequence see A177989.

Numbers n such that the quartic curve y^2=x^4+n doesn't have integral points. see A177987.

Cf. A054504, A081121, A177986, A177987, A177988, A177989.

Magma
```
IntegralQuarticPoints([1,0,0,0,-79]);
```

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]

cp's OEIS Frontend

A177986 Numbers n such that quartic curve y^2=x^4+n have integral points.

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A177987 Numbers n such that quartic equation y^2=x^4+n has no solution.

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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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A080762 Positive numbers not of the form y^2 - x^3, x and y >= 1.

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PARI

A125643 Squares and cubes (with repetition).

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A177988 Numbers n such that quartic curve y^2=x^4-n has integral points.

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Magma

A177989 Numbers n such that quartic equation y^2=x^4-n has no integer solution.

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Magma

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