Laurenţiu Panaitopol

A resultant condition for the irreducibility of the polynomials

Abstract We prove a criterion for the irreducibility of the polynomials in one indeterminate with the coefficients in the valuation ring of a discrete valued field. From this result we deduce the Schonemann, Eisenstein, and Akira irreducibility criteria. The results obtained can also be used for proving that some polynomials in several indeterminates are irreducible.

ON THE RATIO OF CONSECUTIVE PRIMES

For n ≥ 1, let pn be the nth prime number and for n ≥ 1. Using several results of Erdős, we study the sequence (qn)n ≥ 1 and we prove similar results as for the sequence (dn)n ≥ 1, dn = pn+1 - pn. We also consider the sequence for n ≥ 1 and denote by Gn and An its geometrical and arithmetical averages. We prove that for n ≥ 4.

Bounds for Heights of Integer Polynomial Factors

We describe new methods for the estimation of the bounds of the coefficients of proper divisors of integer polynomials in one variable. There exist classes of polynomials for which our estimates are better than those obtained using the polynomial measure or the 2-weighted norm.