A086892 Greatest common divisor of 2^n-1 and 3^n-1.

1, 1, 1, 5, 1, 7, 1, 5, 1, 11, 23, 455, 1, 1, 1, 85, 1, 133, 1, 275, 1, 23, 47, 455, 1, 1, 1, 145, 1, 2387, 1, 85, 23, 1, 71, 23350145, 1, 1, 1, 11275, 1, 2107, 431, 115, 1, 47, 1, 750295, 1, 11, 1, 265, 1, 133, 23, 145, 1, 59, 1, 47322275, 1, 1, 1, 85, 1, 10787, 1, 5, 47, 781, 1

Offset: 1

Author

Joseph H. Silverman (jhs(AT)math.brown.edu), Sep 18 2003

Keywords

• easy
• nonn

Comments

a(n) is a simple (the simplest?) example of a divisibility sequence associated to a rational point on an algebraic group of dimension larger than two. Specifically, it is the divisibility sequence associated to the point (2,3) on the two-dimensional torus G_m^2. Ailon and Rudnick conjecture that a(n) = 1 for infinitely many n.

According to Corvaja, a(n) < 2^n - 1 for all but finitely many n.

References

• Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1. Math. Z. 243 (2003), no. 1, 79-84

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• N. Ailon, Z. Rudnick, Torsion points on curves and common divisors of a^k-1 and b^k-1, Acta Arith. 113 (2004), no. 1, 31-38.
• Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1, Math. Z. 243 (2003), no. 1, 79-84
• P. Corvaja, Greatest Common Divisors in Vojta's Conjecture: Arithmetic and Geometry, Journées Arithmétiques 2011.
• Index to divisibility sequences

Crossrefs

Cf. A000225, A000255, A003462, A024023, A260119.

Programs

Haskell

a086892 n = a086892_list !! (n-1)
a086892_list = tail $ zipWith gcd a000225_list a003462_list
-- Reinhard Zumkeller, Jul 18 2015

Magma

[Gcd(2^n-1, 3^n-1): n in [1..75]]; // Vincenzo Librandi, Sep 02 2015

Maple

seq(igcd(2^n-1,3^n-1), n=1..100); # Robert Israel, Sep 02 2015

Mathematica

Table[GCD[2^n - 1, 3^n - 1], {n, 100}] (* Vincenzo Librandi, Sep 02 2015 *)

PARI

vector(100,n,gcd(2^n-1,3^n-1))

Formula

a(n) = gcd(2^n - 1, 3^n - 1).

a(n) = GCD(A000255(n), A003462(n)) = GCD(A000255(n), A024023(n)). - Reinhard Zumkeller, Mar 26 2004

Extensions

Replaced arXiv URL with non-cached version by R. J. Mathar, Oct 23 2009