Abstract
Let S = {x 1, ..., x n } be a set of n distinct positive integers and e ⩾ 1 an integer. Denote the n × n power GCD (resp. power LCM) matrix on S having the e-th power of the greatest common divisor (x i , x j ) (resp. the e-th power of the least common multiple [x i , x j ]) as the (i, j)-entry of the matrix by ((x i , x j )e) (resp. ([x i , x j ]e)). We call the set S an odd gcd closed (resp. odd lcm closed) set if every element in S is an odd number and (x i , x j ) ∈ S (resp. [x i , x j ] ∈ S) for all 1 ⩽ i, j ⩽ n. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer e ⩾ 1, the n × n power GCD matrix ((x i , x j )e) defined on an odd-gcd-closed (resp. odd-lcm-closed) set S divides the n × n power LCM matrix ([x i , x j ]e) defined on S in the ring M n (ℤ) of n × n matrices over integers. In this paper, we use Hong’s method developed in his previous papers [J. Algebra 218 (1999) 216–228; 281 (2004) 1–14, Acta Arith. 111 (2004), 165–177 and J. Number Theory 113 (2005), 1–9] to investigate Hong’s conjectures. We show that the conjectures of Hong are true for n ⩽ 3 but they are both not true for n ⩾ 4.
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Research is partially supported by Program for New Century Excellent Talents in University, by SRF for ROCS, SEM, China and by the Lady Davis Fellowship at the Technion, Israel
Research is partially supported by a UGC (HK) grant 2160210 (2003/05).
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Zhao, J., Hong, S., Liao, Q. et al. On the divisibility of power LCM matrices by power GCD matrices. Czech Math J 57, 115–125 (2007). https://doi.org/10.1007/s10587-007-0048-6
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DOI: https://doi.org/10.1007/s10587-007-0048-6