| We explore two specific connections between prime numbers and polynomials.;Cohn's Criterion states that if dndn --1 ... d0 is the base 10 representation of a prime, then the polynomial j=0n djxj is irreducible. Let f(x) be a polynomial with non-negative integer coefficients. We define c(10) to be the largest integer such that if f(10) is prime and all the coefficients of f( x) are ≤ c(10), then f( x) is irreducible. It is known that 2.52x1030≤c10 ≤4.96x1031. We improve the lower bound above to 5.21 x 10 30. Furthermore, we classify all reducible polynomials f(x) with non-negative integer coefficients such that f(10) is prime and all the coefficients of f(x) are ≤ 4.96 x 1031.;Let S be a finite set of rational primes. For a non-zero integer n, define [n]S = pip∈S &vbm0;n&vbm0;-1p , where |n|p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [ n(n + 1)...(n + k )]S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]S and [n2 + 7]S, where S = {2, 3} and S = {2}, respectively. We extend Stewart's theorem to prove effective upper bounds for [f(n)]S for an arbitrary f(x) in Z [x] having at least two distinct roots. |
