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In this paper, the authors continue their work on the problem of nding an h = h(x) as small as possible such that for x su ciently large, there is a squarefree number in the interval (x; x + h]: This problem has been investigated by Fogels [4], Roth [11], Richert [10], Rankin [9], Schmidt [12], Graham and Kolesnik [5], the second author [14,15], and the rst… (More)

- Michael Filaseta
- 1991

is irreducible. Irreducibility here and throughout this paper refers to irreducibility over the rationals. Some condition, such as ja0j = janj = 1, on the integers aj is necessary; otherwise, the irreducibility of all polynomials of the form above would imply every polynomial inZ[x] is irreducible (which is clearly not the case). In this paper, we will… (More)

We address conjectures of P. Erdős and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpiński number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k, k2, k3, . . . , kr… (More)

- Michael Filaseta
- 1995

- Michael Filaseta, Andrzej Schinzel
- Math. Comput.
- 2004

An algorithm is described that determines whether a given polynomial with integer coefficients has a cyclotomic factor. The algorithm is intended to be used for sparse polynomials given as a sequence of coefficientexponent pairs. A running analysis shows that, for a fixed number of nonzero terms, the algorithm runs in polynomial time.

in which cases either f(x) is irreducible or f(x) is the product of two irreducible polynomials of equal degree. If |an| = n > 1, then for some choice of a1, . . . , an−1 ∈ Z and a0 = ±1, we have that f(x) is reducible. I. Schur (in [8]) obtained this result in the special case that an = ±1. Further results along the nature of Theorem 1 are also discussed… (More)

Let and N be positive real numbers, and let f : R 7! R be any function. In this paper, we will obtain estimates for the size of the set fu 2 (N; 2N ] : jjf(u)jj < g, where u represents an integer and jjf(u)jj represents the distance from f(u) to the nearest integer. Thus, the set consists of u 2 (N; 2N ] for which ff(u)g, the fractional part of f(u), lies… (More)

- Michael Filaseta
- J. Comb. Theory, Ser. A
- 1985

- Michael Filaseta
- 1998

Throughout this paper, we refer to the non-cyclotomic part of a polynomial f(x) 2 Z[x] as f(x) with its cyclotomic factors removed. More speci cally, if g1(x); : : : ; gr(x) are non-cyclotomic irreducible polynomials in Z[x] and gr+1(x); : : : ; gs(x) are cyclotomic polynomials such that f(x) = g1(x) gr(x) gr+1(x) gs(x), then g1(x) gr(x) is the… (More)

- Michael Filaseta
- 1993