Polynomials with divisors of every degree

2011/11/23 by Thompson, Lola
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1111.5401

Abstract

We consider polynomials of the form tⁿ-1 and determine when members of this family have a divisor of every degree in Z[t]. With F(x) defined to be the number of such integers up to x, we prove the existence of two positive constants c₁ and c₂ such that c₁ x/(log x) ≤ F(x) ≤ c₂ x/(log x).

Related

• Variations on a question concerning the degrees of divisors of xⁿ-1
• On the divisors of xⁿ-1 in F_p[x]
• On the degrees of divisors of Tⁿ-1
• On integers n for which Xⁿ-1 has a divisor of every degree
• Common divisors of totients of polynomial sequences