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 GF = {F(n) } oon = 1. Then F(n) is called composite in GF if F(n) is the product of strictly smaller terms of GF. Otherwise F(n) is prime in GF. It is proved that, if F(x) is not of the form a(bx+c)d, then almost all members of GF are prime in GF. More precisely, if C(x) denotes the number of composite F(n) in GF, with n \geq x, then, for any \epsilon > 0, it is shown that C(x) << x1-(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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