Polynomials with No Small Prime Values

Let /(x) be a polynomial with integer coefficients, and let D(/)-gx.d.{/.(*):*eZ}. It was conjectured by Bouniakowsky in 1857 that if f(x) is nonconstant and irreducible over Z, theii \f(x)\/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 \f(x)\ is prime is large as a function of the degree of / and the size of the coefficients of /. Let f(x) be a polynomial with integer coefficients, and let £>(/) be the largest integer D such that D divides \f{x)\ for all integers x. It was conjectured by Bouniakowsky [B] in 1857 that if f(x) is nonconstant and irreducible over the rationals, then \f(x)\/D(f) is prime for infinitely many integers x. This conjecture is only known to be true in the case where f(x) is of degree one, when Bouniakowsky's conjecture is equivalent to the well-known theorem of Dirichlet on primes in arithmetic progressions. If Bouniakowsky's conjecture is true, then it seems natural to ask the question: How large is the smallest integer x for which \f(x)\/D(f) is prime? In the case where f(x) is of degree one, an answer to this question is provided by a result of Linnik, that if (a,q) = 1, then the least prime congruent to a modulo q does not exceed qc>. (In this note we use cx,c2,... to denote positive absolute constants.) On the other hand, it was proved by Prachar [P] that there exist positive integers a and q with a < q and {a, q) = 1 such that a + qx is composite for all integers x with 0 < x < c2log<7log2<7-—-, where logkq is the /c-fold iterated natural logarithm. In a previous paper by the author [M] a result was proved for polynomials of higher degree that is analogous to the result of Prachar. The purpose of the present note is to prove a stronger result of this type. In order to provide a means to measure the size of the least x for which \f(x)\/D(f) is prime, we define the length L(f) of a polynomial as follows. Definition. If f{x) = Y."k=0akxk with ak e Z, then L(f) = ££_0llaJI» where ||aA.|| is the number of digits in the binary expansion of ak, with ||0|| = 1. Received by the editors March 28, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 11N32, 11R09. >i'1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 393 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use