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Integer sequences defined by linear recurrences have attracted a great deal of study, particulary binary linear recurrences such as the Fibonacci and Lucas sequences. A natural question to ask about such sequences is how close their terms are to being prime. One way of studying this is to examine the set of primes dividing at least one term of the sequence, denoted P(a_n). The sparser this set is, the closer the terms are to being prime in some sense. In the case of binary linear recurrences, the density of P(a_n) is known in many cases, and I'll review some of these results. Then I'll discuss non-linear recurrences arising from iteration of a quadratic polynomial. In general much less is known about these recurrences, but the density of P(a_n) can be studied via the Galois theory of iterates of the polynomial. I will explain this connection and prove results in some cases. At the end of the talk I'll discuss several open questions that arise regarding specific polynomials, like the nefarious x^2 - 1.
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