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- Title
On Nonsingular Power LCM Matrices.
- Authors
Hong, Shaofang; Shum, K. P.; Sun, Qi
- Abstract
Let e ≥ 1 be an integer and S={x1,...,xn} a set of n distinct positive integers. The matrix ([xi, xj]e) having the power [xi, xj]e of the least common multiple of xi and xj as its (i, j)-entry is called the power least common multiple (LCM) matrix defined on S. The set S is called gcd-closed if (xi,xj) ∈ S for 1≤ i, j≤ n. Hong in 2004 showed that if the set S is gcd-closed such that every element of S has at most two distinct prime factors, then the power LCM matrix on S is nonsingular. In this paper, we use Hong's method developed in his previous papers to consider the next case. We prove that if every element of an arbitrary gcd-closed set S is of the form pqr, or p2qr, or p3qr, where p, q and r are distinct primes, then except for the case e=1 and 270, 520 ∈ S, the power LCM matrix on S is nonsingular. We also show that if S is a gcd-closed set satisfying xi< 180 for all 1≤ i≤ n, then the power LCM matrix on S is nonsingular. This proves that 180 is the least primitive singular number. For the lcm-closed case, we establish similar results.
- Subjects
NUMBER theory; ALGEBRAIC number theory; LINEAR algebra; EIGENVALUES; ARITHMETIC functions; FACTORIZATION
- Publication
Algebra Colloquium, 2006, Vol 13, Issue 4, p689
- ISSN
1005-3867
- Publication type
Article
- DOI
10.1142/S1005386706000642