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Factoring polynomial values

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  • Adam
    Let p(x) be any polynomial with integer coefficients and degree 0. Is the set {p:p a prime, p divides p(i) for some integer i} infinite? Adam
    Message 1 of 2 , Mar 22, 2003
      Let p(x) be any polynomial with integer coefficients and degree>0.
      Is the set {p:p a prime, p divides p(i) for some integer i} infinite?

      Adam
    • Satoshi Tomabechi
      On Sat, 22 Mar 2003 18:23:52 -0000 ... Yes, it is. Hasse-Weil inequality |a(p)-p-1| 0, where g is the genus of
      Message 2 of 2 , Mar 23, 2003
        On Sat, 22 Mar 2003 18:23:52 -0000
        "Adam" <a_math_guy@...> wrote:

        > Let p(x) be any polynomial with integer coefficients and degree>0.
        > Is the set {p:p a prime, p divides p(i) for some integer i} infinite?

        Yes, it is.
        Hasse-Weil inequality |a(p)-p-1|<=2*g*sqrt(p) shows that
        a(p)!=0 when p >> 0,
        where g is the genus of algebraic curve defined by p(x)
        and a(p) is the number of roots of p(x) = 0 mod p.

        Satoshi Tomabechi
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