Computational experiences on the distances of polynomials to irreducible polynomials

In this paper we deal with a problem of Turan concerning the 'distance' of polynomials to irreducible polynomials. Using computational methods we prove that for any monic polynomial P epsilon Z[x] of degree less than or equal to 22 there exists a monic polynomial Q epsilon Z[x] with deg(Q) = deg(P) such that Q is irreducible over Q and the 'distance' of P and Q is less than or equal to 4.