2013/10/09 by Andrzej Nowicki, Nowicki, Andrzej
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1310.2416
arxiv created 2013/10/09 · arxiv updated 2013/10/10
Let R be a gcd-domain (for example let R be a unique factorization domain), and let (an)n\geqslant1 be a sequence of nonzero elements in R. We prove that gcd(an,am)=agcd(n,m) for all n,m\geqslant1 if and only if an=∏d| n cd\quadfor n\geqslant1, where c1=a1 and cn=lcm(a1,a2,…,an)/lcm(a1,a2,…,an-1) for n\geqslant2. All equalities with gcd and lcm are determined up to units of R.