New York Journal of Mathematics
Volume 19 (2013) 91-116

  

Paul Pollack and Lola Thompson

On the degrees of divisors of Tn-1

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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 DF(n) ⊂ [0,n] defined by
DF(n) := {0 ≦ m ≦ n : Tn-1 has a divisor of degree m in F[T]}.
When DF(n) consists of all integers m with 0 ≦ m ≦ n, so that Tn-1 has a divisor of every degree, we call n an F-practical 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,
#{F-practical n≦ x} \asympF (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 DF(n)? We prove upper bounds in this problem for both F=Q and F=Fp (with p prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such n ≦ x is OF (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