Glasnik Matematicki, Vol. 52, No. 2 (2017), 235-240.

ROOTS OF UNITY AS QUOTIENTS OF TWO CONJUGATE ALGEBRAIC NUMBERS

Artūras Dubickas

Abstract.   Let α be an algebraic number of degree d ≥ 2 over Q. Suppose for some pairwise coprime positive integers n₁,… ,n_r we have deg(α^n_j) < d for j=1,…,r, where deg(αⁿ)=d for each positive proper divisor n of n_j. We prove that then φ(n₁ … n_r) ≤ d, where φ stands for the Euler totient function. In particular, if n_j=p_j, j=1,…,r, are any r distinct primes satisfying deg(α^p_j) < d, then the inequality (p₁-1)… (p_r-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.

2010 Mathematics Subject Classification.   11R04, 11R18.

Key words and phrases.   Root of unity, conjugate algebraic numbers, degenerate linear recurrence sequence.

DOI: 10.3336/gm.52.2.03

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