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Prime factors of dynamical sequences
-
Xander Faber
and
Andrew Granville
Published/Copyright:
December 1, 2011
Abstract
Let (t) (t) have degree d 2. For a given rational number x0, define xn1(xn) for each n 0. If this sequence is not eventually periodic, and if does not lie in one of two explicitly determined affine conjugacy classes of rational functions, then xn1xn has a primitive prime factor in its numerator for all sufficiently large n. The same result holds for the exceptional maps provided that one looks for primitive prime factors in the denominator of xn1xn. Hence the result for each rational function of degree at least 2 implies (a new proof) that there are infinitely many primes. The question of primitive prime factors of xnxn is also discussed for uniformly bounded.
Received: 2009-10-21
Revised: 2010-07-18
Published Online: 2011-December
Published in Print: 2011-December
Walter de Gruyter Berlin New York 2011
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Articles in the same Issue
- On the cyclotomic main conjecture for the prime 2
- Some consequences of Arthur's conjectures for special orthogonal even groups
- Cartier modules: Finiteness results
- RiemannHilbert problem for Hurwitz Frobenius manifolds: Regular singularities
- Prime factors of dynamical sequences
- Enriques manifolds
- Volume versus rank of lattices
Articles in the same Issue
- On the cyclotomic main conjecture for the prime 2
- Some consequences of Arthur's conjectures for special orthogonal even groups
- Cartier modules: Finiteness results
- RiemannHilbert problem for Hurwitz Frobenius manifolds: Regular singularities
- Prime factors of dynamical sequences
- Enriques manifolds
- Volume versus rank of lattices
