Some Estimates Involving Density of Algebraic Numbers and Integer Polynomials

Abstract

Nymann in 1970 derived an asymptotic formula for the probability that k positive integers, chosen at random from the first n natural numbers, are relatively prime. In 1996, Arno et al. introduced a new concept called the denominator of an integer polynomial. Using this concept, Arno et al. proved theorems establishing formulae for determining the denominator of any algebraic number and the density of algebraic numbers whose denominators are equal to the leading coefficients in their minimal polynomials. The proofs of Arno et al. made use of the result of Nymann . The first part of this paper is an extension of the work of Nymann done by relaxing the condition that the chosen numbers are relatively prime. In the second part, the formulae derived in the first part are employed to find asymptotic estimates and the density of the set of integer polynomials refining the work of Arno et al.