 

Paul Pollack and Lola Thompson
On the degrees of divisors of T^{n}1 view print


Published: 
February 23, 2013 
Keywords: 
Multiplicative order; practical number; cyclotomic polynomials 
Subject: 
Primary: 11N25, Secondary: 11N37 


Abstract
Fix a field F. In this paper, we study the sets D_{F}(n) ⊂ [0,n] defined by
D_{F}(n) := {0 ≦ m ≦ n : T^{n}1 has a divisor of degree m in F[T]}.
When D_{F}(n) consists of all integers m with 0 ≦ m ≦ n, so that T^{n}1 has a divisor of every degree, we call n an Fpractical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers n for which every integer 0 ≦ m ≦ σ(n) can be written as a sum of distinct divisors of n. Our first theorem states that, for any number field F and any x ≧ 2,
#{Fpractical n≦ x} \asymp_{F} (x/log x);
this extends work of the second author, who obtained this estimate when F=Q.
Suppose now that x ≧ 3, and let m be a natural number in [3,x]. We ask: For how many n ≦ x does m belong to D_{F}(n)? We prove upper bounds in this problem for both F=Q and F=F_{p} (with p prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such n ≦ x is O_{F} (x/(log m)^{2/35}), uniformly in m.


Author information
Paul Pollack:
Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, Georgia 30602, USA
pollack@uga.edu
Lola Thompson:
Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, Georgia 30602, USA
lola@math.uga.edu

