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arXiv · 0707.2505

Primitive Divisors in Arithmetic Dynamics

Patrick Ingram·Joseph H. Silverman
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Abstract

Let F(z) be a rational function in Q(z) of degree at least 2 with F(0) = 0 and such that F does not vanish to order d at 0. Let b be a rational number having infinite orbit under iteration of F, and write F^n(b) = A_n/B_n as a fraction in lowest terms. We prove that for all but finitely many n > 0, the numerator A_n has a primitive divisor, i.e., there is a prime p such that p divides A_n and p does not divide A_i for all i < n. More generally, we prove an analogous result when F is defined over a number field and 0 is a periodic point for F.

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Patrick Ingram, Joseph H. Silverman. 2007-07-17. Primitive Divisors in Arithmetic Dynamics. https://doi.org/10.1017/s0305004108001795

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