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E9. A prime generating function - not

A student claims to have found a quadratic function f(n) = an^2 + bn + c, with a, b, and c integers, such that f(n) is prime for all positive integers n. Disprove the claim by showing that such a function always takes on a composite value. This is a special case of a well-known result for general polynomial functions. What is sought here is a proof that uses no more than high school algebra.

Hints. Without loss of generality, a > 0. If c = 0 the result is trivial and for |c| > 1, the result is clear, since c divides f(c), so it suffices to prove the result for |c| = 1.

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