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Thread: Prime polynomial

  1. #1
    Senior Member
    Apr 2009

    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
    Senior Member
    Nov 2007
    No polynomial $\displaystyle f (n)$ with integral coefficients, not a constant, can be prime for all $\displaystyle n$, or for all sufficiently large $\displaystyle n$.
    We may assume that the leading coefficient in $\displaystyle f(n)$ is positive, so that $\displaystyle f(n)\rightarrow\infty$ when $\displaystyle n\rightarrow\infty$, and $\displaystyle f(n) > 1$ for $\displaystyle n > N$, say. If $\displaystyle x > N$ and $\displaystyle f(x)=a_0x^k+\ldots =y>1$ then $\displaystyle f(ry+x)=a_0\left(ry+x\right)^k+\ldots$ is divisible by $\displaystyle y$ for every integral $\displaystyle r$; and $\displaystyle f(ry+x)$ tends to infinity with $\displaystyle r$. Hence there are infinitely many composite values of $\displaystyle f(n)$.
    Last edited by james_bond; Jan 1st 2010 at 01:05 AM.
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