%0 Journal Article
%T Roots of unity as quotients of two conjugate algebraic numbers
%A Dubickas
%A Art¨±ras
%J -
%D 2017
%R 10.3336/gm.52.2.03
%X Sa£¿etak Let ¦Á be an algebraic number of degree d ¡Ý 2 over Q. Suppose for some pairwise coprime positive integers n1,¡­ ,nr we have deg(¦Ánj) < d for j=1,¡­,r, where deg(¦Án)=d for each positive proper divisor n of nj. We prove that then ¦Õ(n1 ¡­ nr) ¡Ü d, where ¦Õ stands for the Euler totient function. In particular, if nj=pj, j=1,¡­,r, are any r distinct primes satisfying deg(¦Ápj) < d, then the inequality (p1-1)¡­ (pr-1) ¡Ü d holds, and therefore r £¿ log d/log log d for d ¡Ý 3. This bound on r improves that of Dobrowolski r ¡Ü log d/log 2 proved in 1979 and is best possible
%K Root of unity
%K conjugate algebraic numbers
%K degenerate linear recurrence sequence
%U https://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=278921