科研之友
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摘要
Let e and n be positive integers and S = {x(1), . . . , x(n)} a set of n distinct positive integers. For x is an element of S, define G(S)(x) := {d is an element of S vertical bar d < x, d vertical bar x and (d vertical bar y vertical bar x, Y is an element of S) double right arrow Y is an element of {d, x}}. The n x n matrix whose (i, j)-entry is the eth power (x(i), x(j))(e) of the greatest common divisor of x(i) and xj is called the eth power GCD matrix on S, denoted by (S-e). Similarly we can define the eth power LCM matrix vertical bar S-e vertical bar. Bourque and Ligh showed that (S) vertical bar vertical bar S vertical bar holds in the ring of n x n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with vertical bar S vertical bar <= 3, (S) vertical bar vertical bar S vertical bar. Meanwhile Hong proved that there is a gcd-closed set S with max(x is an element of S) {vertical bar G(S)(x)vertical bar} = 2 such that (S) vertical bar vertical bar S vertical bar. In this paper, we introduce a new method to study systematically the divisibility for the case max(x is an element of S){vertical bar G(S)(x)vertical bar} <= 2. We give a new proof of Hong's conjecture and obtain necessary and Sufficient conditions on the gcd-closed set S with max(x is an element of S) {vertical bar G(S)(x)vertical bar} = 2 such that (S-e)vertical bar vertical bar S-e vertical bar. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.
