A103255 Integers x > 0 such that x^3 + y^3 = z^2 for some y > 0, z > 0, and gcd(x,y) = 1.

1, 2, 11, 23, 37, 56, 57, 65, 112, 122, 193, 217, 242, 305, 312, 433, 592, 781, 851, 877, 889, 913, 1001, 1064, 1177, 1201, 1346, 1376, 1617, 1633, 1706, 1729, 1801, 1953, 1960, 1969, 2137, 2162, 2184, 2257, 2345, 2480, 2543, 2920, 3071, 3081, 3482, 3641, 3889, 4019

Offset: 1

Author

Cino Hilliard, Mar 20 2005

Keywords

• nonn
• more

Examples

			x=11, y=37, 11^3 + 37^3 = 228^2. 11 is the third entry in the list.
The pairs [x,y] = [a(n),a(?)] for the first few terms are [1, 2], [2, 1], [11, 37], [23, 1177], [37, 11], [56, 65], [57, 112], [65, 56], [112, 57], [122, 1201], [193, 3482], [217, 312], [242, 433]. [_Joerg Arndt_, Sep 30 2012]
		

Links

• F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
• H. Darmon and A. Granville, On the equations z^m=F(x,y) and Ax^p+By^q=Cz^r, Bull. Lond. Math. Soc., 27 (6) (1995) 513, Sect 7.2.

Crossrefs

Cf. A099426 (values of z).

Programs

Magma

[ k : k in [1..100] | exists{P : P in IntegralPoints(EllipticCurve([0,k^3])) | P[1] gt 0 and P[2] ne 0 and GCD(Integers()!P[1],k) eq 1} ]; // Geoff Bailey

Mathematica

(* This program uses z-values from A099426 b-file. To get 50 terms, the first 200 z-values suffice, the result being the same with the whole b-file of 300 z-values. *)
terms = 50;
zz = Import["https://oeis.org/A099426/b099426.txt", "Table"][[1 ;; 4 terms, 2]];
r[z_] := {x, y, z} /. ToRules[Reduce[GCD[x, y] == 1 && 0Jean-François Alcover, Jun 13 2019 *)

PARI

is_A103255(x, lim)=
{ /* Warning: just how big lim has to be is unclear */
    my(x3=x^3);
    for (y=1, lim,
        if ( gcd(x,y) != 1, next() );
        if ( issquare(x3+y^3), return(1) );
    );
    return(0);
}
/* Using lim=10^6 reproduces all terms <= 1000: */
for (n=1,1000, if( is_A103255(n, 10^6), print1(n,", ")) );
/* Joerg Arndt, Sep 30 2012 */

Sage

# apparently inefficient as of version 5.2
def is_A103255(n):
    E = EllipticCurve([0, n^3])
    E.gens(descent_second_limit=16);
    for p in E.integral_points():
        if p[0] > 0 and p[1] > 0 and gcd(p[1], n) == 1:
            return true
    return false
[n for n in (1..60) if is_A103255(n)]
# Peter Luschny, Sep 29 2012

Extensions

Recomputed and extended by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using MAGMA, Jan 28 2007

a(9)-a(10) from Jonathan Vos Post, May 27 2007

a(11)-a(16) from Vincenzo Librandi, Dec 21 2010

a(17)-a(22) from Joerg Arndt, Sep 30 2012

a(23)-a(50) from Jean-François Alcover, Jun 12 2019