# Irreducible polynomials in Z [x]

#### Abstract

Any irreducible polynomial f(x) in Z [x] such that the set of values f( Z^{+} ) has no common divisor larger than 1 represents prime numbers infinitely often. The idea is to produce prime numbers from irreducible polynomials. The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 to the effect that any irreducible polynomial f(x) in Z [x] such that the set of values f( Z^{+} ) has no common divisor larger than 1 represents prime numbers infinitely often. This conjecture is still open and one of the major unsolved problems in number theory when the degree of f is greater than one. When the degree of f is one (i.e., when f is linear), the result is true following Dirichlet's theorem on primes in arithmetic progressions. ^ In other words the irreducible polynomial *f*( x) = *k* x + *l* ∈ Z [x] where *k* and *l* are fixed relatively prime integers, and x takes on the values 0, 1, 2… contains infinitely many primes. ^

#### Subject Area

Mathematics

#### Recommended Citation

Getnet Abebe Gidelew,
"Irreducible polynomials in Z [x]"
(2008).
*ETD Collection for Tennessee State University.*
Paper AAI1455166.

https://digitalscholarship.tnstate.edu/dissertations/AAI1455166