On the infinitude of prime divisors of the range of a non-constant polynomial
by
Ivan Gotchev
Central Connecticut State University

Let Z be the set of all integers, N be the set of all positive integers, n in N and P_n(x)=a_nxⁿ+a_n-1x^n-1+...+a₀ be a polynomial with integer coefficients. Also, let S={ P_n(z)|z in Z }\{ 0} and let G be the greatest common divisor of the elements of S.

In 1857 V. Bouniakowsky [1] formulated the following conjecture, which is still open.

If P_n(x) is a non-constant irreducible polynomial with integer coefficients, then the polynomial P_n(x)/G takes prime number values for infinitely many integers x (see also [4]).

In order for this conjecture to be true, it is necessary for the elements of S to have infinitely many distinct prime divisors. In fact, this weaker condition is true for any non-constant polynomial with integer coefficients.

Theorem 1. Let P_n(x)=a_nxⁿ+a_n-1x^n-1+...+a₀, n in N be a non-constant polynomial with integer coefficients and let S={ P_n(z)|z in Z }\{ 0}. Then the elements of S have infinitely many distinct prime divisors.

Following Furstenberg's topological proof of the fact that there are infinitely many prime numbers [2], we give a topological proof of the above theorem, which answers a question of Ali Özluk [3].

References

[1] V. Bouniakowsky, Sur les diviseurs numériques invariables des fonctions rationnelles entrières, Acad. Sci. St. Pétersbourg Mém., sci. math. et phys. 6 (1857) 305-329.

[2] H. Furstenberg, On the infinitude of primes, Amer. Math. Monthly 62 (1955) 353.

[3] A. Özluk, Private communication.

[4] W. Sierpi\'nski, Elementary theory of numbers, PWN, Warszawa 1964.

Date received: May 31, 2003