# Article

Full entry | PDF   (0.3 MB)
Keywords:
gcd-closed set; greatest-type divisor(GTD); maximal gcd-fixed set(MGFS); least common multiple matrix; power LCM matrix; nonsingularity
Summary:
A set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace$ of $n$ distinct positive integers is said to be gcd-closed if $(x_{i},x_{j})\in \mathcal{S}$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k(t)$ depending only on $t$, such that if $n\le k(t)$, then the power LCM matrix $([x_i,x_j]^t)$ defined on any gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace$ is nonsingular, but for $n\ge k(t)+1$, there exists a gcd-closed set $\mathcal{S}=\lbrace x_1,\ldots ,x_n\rbrace$ such that the power LCM matrix $([x_i,x_j]^t)$ on $\mathcal{S}$ is singular. In 1996, Hong proved $k(1)=7$ and noted $k(t)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that $k(t)\ge 8$ for all $t\ge 2$. We further prove that $k(t)\ge 9$ iff a special Diophantine equation, which we call the LCM equation, has no $t$-th power solution and conjecture that $k(t)=8$ for all $t\ge 2$, namely, the LCM equation has $t$-th power solution for all $t\ge 2$.
References:
[1] S. Beslin: Reciprocal GCD matrices and LCM matrices. Fibonacci Quart. 29 (1991), 271–274. MR 1114893 | Zbl 0738.11026
[2] S. Beslin and S. Ligh: Greatest common divisor matrices. Linear Algebra Appl. 118 (1989), 69–76. DOI 10.1016/0024-3795(89)90572-7 | MR 0995366
[3] K. Bourque and S. Ligh: Matrices associated with classes of arithmetical functions. J. Number Theory 45 (1993), 367–376. DOI 10.1006/jnth.1993.1083 | MR 1247390
[4] K. Bourque and S. Ligh: On GCD and LCM matrices. Linear Algebra Appl. 174 (1992), 65–74. DOI 10.1016/0024-3795(92)90042-9 | MR 1176451
[5] K. Bourque and S. Ligh: Matrices associated with classes of multiplicative functions. Linear Algebra Appl. 216 (1995), 267–275. MR 1319990
[6] S. Z. Chun: GCD and LCM power matrices. Fibonacci Quart. 34 (1996), 290–297. MR 1394756
[7] P. Haukkanen, J. Wang and J. Sillanpää: On Smith’s determinant. Linear Algebra Appl. 258 (1997), 251–269. MR 1444107
[8] S. Hong: LCM matrix on an r-fold gcd-closed set. J. Sichuan Univ. Nat. Sci. Ed. 33 (1996), 650–657. MR 1440627 | Zbl 0869.11021
[9] S. Hong: On Bourque-Ligh conjecture of LCM matrices. Adv. in Math. (China) 25 (1996), 566–568. MR 1453166 | Zbl 0869.11022
[10] S. Hong: On LCM matrices on GCD-closed sets. Southeast Asian Bull. Math. 22 (1998), 381–384. MR 1811182 | Zbl 0936.15011
[11] S. Hong: On the Bourque-Ligh conjecture of least common multiple matrices. J. Algebra 218 (1999), 216–228. DOI 10.1006/jabr.1998.7844 | MR 1704684 | Zbl 1015.11007
[12] S. Hong: Gcd-closed sets and determinants of matrices associated with arithmetical functions. Acta Arith. 101 (2002), 321–332. DOI 10.4064/aa101-4-2 | MR 1880046 | Zbl 0987.11014
[13] S. Hong: On the factorization of LCM matrices on gcd-closed sets. Linear Algebra Appl. 345 (2002), 225–233. MR 1883274 | Zbl 0995.15006
[14] S. Hong: Notes on power LCM matrices. Acta Arith. 111 (2004), 165–177. DOI 10.4064/aa111-2-5 | MR 2039420 | Zbl 1047.11022
[15] S. Hong: Nonsingularity of matrices associated with classes of arithmetical functions. J.  Algebra 281 (2004), 1–14. DOI 10.1016/j.jalgebra.2004.07.026 | MR 2091959 | Zbl 1064.11024
[16] S. Hong: Nonsingularity of least common multiple matrices on gcd-closed sets. J. Number Theory 113 (2005), 1–9. DOI 10.1016/j.jnt.2005.03.004 | MR 2141756 | Zbl 1080.11022
[17] H. J. S. Smith: On the value of a certain arithmetical determinant. Proc. London Math. Soc. 7 (1875–1876), 2080–212.

Partner of