Irreducible polynomials in Z [x]
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. ^
Getnet Abebe Gidelew,
"Irreducible polynomials in Z [x]"
ETD Collection for Tennessee State University.