2017/11/18 by Yuta Suzuki, Suzuki, Yuta
Mathematics · #11A25 #11J25 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A25 #msc:11J25
paper · pdf · doi:10.48550/arxiv.1711.06847
23 pages; C program source code is included with source documents
arxiv created 2017/11/18 · arxiv updated 2017/11/21
For an integer k≥2, a tuple of k positive integers (Mi)i=1k is called an amicable k-tuple if the equation σ(M1)=⋯=σ(Mk)=M1+⋯+Mk holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (1917) conjectured that there is no relatively prime amicable pairs and Artjuhov (1975) and Borho (1974) proved that for any fixed positive integer K, there are only finitely many relatively prime amicable pairs (M,N) with ω(MN)=K. Recently, Pollack (2015) obtained an upper bound MN<(2K)^2K2 for such amicable pairs. In this paper, we improve this upper bound to MN<(π2)/(6)24K-2⋅ 2K and generalize this bound to some class of general amicable tuples.