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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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