2015/10/29 by Ercan Altınışık, Altınışık, Ercan, Mehmet Yıldız +3 · 3 citations
Mathematics · #11A25 #11C20 #15B36 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A25 #msc:11C20 #msc:15B36
paper · pdf · doi:10.48550/arxiv.1510.09101
20 pages, 1 figure
arxiv created 2015/10/29 · arxiv updated 2015/11/02
In this paper, we consider the divisibility problem of LCM matrices by GCD matrices in the ring Mn(ℤ) proposed by Hong in 2002 and in particular a conjecture concerning the divisibility problem raised by Zhao in 2014. We present some certain gcd-closed sets on which the LCM matrix is not divisible by the GCD matrix in the ring Mn(ℤ). This could be the first theoretical evidence that Zhao's conjecture might be true. Furthermore, we give the necessary and sufficient conditions on the gcd-closed set S with |S|≤ 8 such that the GCD matrix divides the LCM matrix in the ring Mn(ℤ) and hence we partially solve Hong's problem. Finally, we conclude with a new conjecture that can be thought as a generalization of Zhao's conjecture.