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Primitive divisors in arithmetic dynamics. (English) Zbl 1242.11012
Summary: Let \(\varphi(z)\in \mathbb Q (z)\) be a rational function of degree \(d \geq 2\) with \(\varphi(0) = 0\) and such that \(\varphi\) does not vanish to order \(d\) at \(0\). Let \(\alpha \in \mathbb Q \) have infinite orbit under iteration of \(\varphi\) and write \(\varphi^n(\alpha) = A_n/B_n\) as a fraction in lowest terms. We prove that for all but finitely many \(n \geq 0\), the numerator \(A_n\) has a primitive divisor, i.e., there is a prime \(p\) such that \(p\mid A_n\) and \(p \nmid A_i\) for all \(i < n\). More generally, we prove an analogous result when \(\varphi\) is defined over a number field and 0 is a preperiodic point for \(\varphi\).

MSC:
11B37 Recurrences
11A41 Primes
37P55 Arithmetic dynamics on general algebraic varieties
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