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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 346.10027

**Autor: ** Cohen, S.D.; Erdös, Paul; Nathanson, M.B.

**Title: ** Prime polynomial sequences. (In English)

**Source: ** J. London Math. Soc., II. Ser. 14, 559-562 (1976).

**Review: ** Let F(x) be a polynomial of degree d \geq 2 with integral coefficients and such that F(n) \geq 1 for all n \geq 1, Let __G___{F} = **{**F(n) **}** ^{oo}_{n = 1}. Then F(n) is called composite in __G___{F} if F(n) is the product of strictly smaller terms of __G___{F}. Otherwise F(n) is prime in __G___{F}. It is proved that, if F(x) is not of the form a(bx+c)^{d}, then almost all members of __G___{F} are prime in __G___{F}. More precisely, if C(x) denotes the number of composite F(n) in __G___{F}, with n \geq x, then, for any \epsilon > 0, it is shown that C(x) << x^{1-(1/d2)+\epsilon}. For monic quadratics an identity implies that C(x) >> x^{ 1/2 } so that in this case x{^{1}/_{2}} << C(x) << x^{ 3/4 +\epsilon}. On the other hand, it is easy to construct polynomials for which C(x) = 0 for all x. In general, the exact order of C(x) is unknown.

**Classif.: ** * 11N13 Primes in progressions

11B83 Special sequences of integers and polynomials

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