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Polynomials with no small prime values


Author: Kevin S. McCurley
Journal: Proc. Amer. Math. Soc. 97 (1986), 393-395
MSC: Primary 11N32; Secondary 11R09
DOI: https://doi.org/10.1090/S0002-9939-1986-0840616-4
MathSciNet review: 840616
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Abstract: Let $ f(x)$ be a polynomial with integer coefficients, and let

$\displaystyle D(f) = {\text{g}}{\text{.c}}{\text{.d}}\{ f(x):x \in {\mathbf{Z}}\}.$

It was conjectured by Bouniakowsky in 1857 that if $ f(x)$ is nonconstant and irreducible over $ {\mathbf{Z}}$, then $ \vert f(x)\vert/D(f)$ is prime for infinitely many integers $ x$. It is shown that there exist irreducible polynomials $ f(x)$ with $ D(f) = 1$ such that the smallest integer $ x$ for which $ \vert f(x)\vert$ is prime is large as a function of the degree of $ f$ and the size of the coefficients of $ f$.

References [Enhancements On Off] (What's this?)

  • [AO] L. Adleman and A. Odlyzko, Irreducibility testing and factorization of polynomials, Math. Comp. 41 (1983), 699-709. MR 717715 (86f:11097)
  • [B] V. Bouniakowsky, Sur les diviseurs numeriques invariables des fonctions rationelles entires, Mem. Acad. Sci. St. Petersburg 6 (1857), 305-329.
  • [M] K. McCurley, Prime values of polynomials and irreducibility testing, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 155-158. MR 741729 (85e:11066)
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DOI: https://doi.org/10.1090/S0002-9939-1986-0840616-4
Article copyright: © Copyright 1986 American Mathematical Society

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