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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomials with no small prime values


Author: Kevin S. McCurley
Journal: Proc. Amer. Math. Soc. 97 (1986), 393-395
MSC: Primary 11N32; Secondary 11R09
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$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0840616-4
PII: S 0002-9939(1986)0840616-4
Article copyright: © Copyright 1986 American Mathematical Society