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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

Sažetak

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.

Ključne riječi

Root of unity; conjugate algebraic numbers; degenerate linear recurrence sequence

Hrčak ID:

189330

URI

https://hrcak.srce.hr/189330

Datum izdavanja:

13.11.2017.

Posjeta: 1.699 *