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Thread: Prime polynomial

  1. #1
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    Prime polynomial

    Prove that there is no non-constant polynomial f(x), with integer coefficients, such that f(x) is prime for all integers x.

    I just started modular arithmetic, so if someone can explain to me how to apply modular arithmetic to prove this, it would be much appreciated!
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  2. #2
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    No polynomial f (n) with integral coefficients, not a constant, can be prime for all n, or for all sufficiently large n.
    We may assume that the leading coefficient in f(n) is positive, so that f(n)\rightarrow\infty when n\rightarrow\infty, and f(n) > 1 for n > N, say. If x > N and f(x)=a_0x^k+\ldots =y>1 then f(ry+x)=a_0\left(ry+x\right)^k+\ldots is divisible by y for every integral r; and f(ry+x) tends to infinity with r. Hence there are infinitely many composite values of f(n).
    Last edited by james_bond; Jan 1st 2010 at 01:05 AM.
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