Glasnik matematički, Vol. 52 No. 2, 2017.
Original scientific paper
https://doi.org/10.3336/gm.52.2.03
Roots of unity as quotients of two conjugate algebraic numbers
Artūras Dubickas
orcid.org/0000-0002-3625-9466
; Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania
Abstract
Let α be an algebraic number of degree d ≥ 2 over Q. Suppose for some pairwise coprime positive integers n1,… ,nr we have deg(αnj) < d for j=1,…,r, where deg(αn)=d for each positive proper divisor n of nj. We prove that then φ(n1 … nr) ≤ d, where φ stands for the Euler totient function. In particular, if nj=pj, j=1,…,r, are any r distinct primes satisfying deg(αpj) < d, then the inequality (p1-1)… (pr-1) ≤ d holds, and therefore r ≪ log d/log log d for d ≥ 3. This bound on r improves that of Dobrowolski r ≤ log d/log 2 proved in 1979 and is best possible.
Keywords
Root of unity; conjugate algebraic numbers; degenerate linear recurrence sequence
Hrčak ID:
189330
URI
Publication date:
13.11.2017.
Visits: 1.699 *