Reducibility of polynomials and covering systems of congruences

Computational experiences on the distances of polynomials to irreducible polynomials

“…In 1962 P. Turán proposed the following problem (cf. [10] This deep theorem gives a partial answer to Turán's problem. A similar problem was proposed in 1984 by M. Szegedy (cf.…”

The distance to an irreducible polynomial, II

“…More than 40 years ago, P. Turán [7] asked if every polynomial with integer coefficients lies near an irreducible polynomial with the same degree or smaller, where distance is measured by using the length. More precisely, he asked if there exists an absolute constant C such that for every polynomial f ∈ Z[x] there exists an irreducible polynomial g ∈ Z[x] with deg(g) ≤ deg(f ) and L(f − g) ≤ C. Note that if such a constant C exists, then certainly C ≥ 2, as this value is required for f (x) = x n when n is odd and n ≥ 3, or for f (x) = x n−2 (x 2 + x − 1) when n is even and n ≥ 4 .…”

On the irreducibility of 0,1-polynomials of the form f(x) xn+ g(x)

“…The work in [2] was motivated by work of Schinzel [3,4] where a similar result is obtained without an explicit estimate on n 0 (though the methods there do allow for such an estimate).…”