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Keywords:
GCD-closed set; LCM-closed set; greatest-type divisor; divisibility
GCD-closed set; LCM-closed set; greatest-type divisor; divisibility
Summary:
Let $S=\lbrace x_1,\dots ,x_n\rbrace $ be a set of $n$ distinct positive integers and $e\ge 1$ an integer. Denote the $n\times n$ 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 $(x_i,x_j)\in S$ (resp. $[x_i, x_j]\in S$) for all $1\le i,j \le n$. In studying the divisibility of the power LCM and power GCD matrices, Hong conjectured in 2004 that for any integer $e\ge 1$, the $n\times n$ power GCD matrix $((x_i, x_j)^e)$ defined on an odd-gcd-closed (resp. odd-lcm-closed) set $S$ divides the $n\times n$ power LCM matrix $([x_i, x_j]^e)$ defined on $S$ in the ring $M_n({\mathbb Z})$ of $n\times n$ 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 $n\le 3$ but they are both not true for $n\ge 4$.
References:
[1] T. M. Apostol: Arithmetical properties of generalized Ramanujan sums. Pacific J. Math. 41 (1972), 281–293. DOI 10.2140/pjm.1972.41.281 | MR 0311597 | Zbl 0226.10045
[2] S. Beslin and S. Ligh: Another generalization of Smith’s determinant. Bull. Austral. Math. Soc. 40 (1989), 413–415. DOI 10.1017/S0004972700017457 | MR 1037636
[3] 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
[4] K. Bourque and S. Ligh: Matrices associated with arithmetical functions. Linear and Multilinear Algebra 34 (1993), 261–267. DOI 10.1080/03081089308818225 | MR 1304611
[5] 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
[6] K. Bourque and S. Ligh: Matrices associated with classes of multiplicative functions. Linear Algebra Appl. 216 (1995), 267–275. MR 1319990
[7] C. He and J. Zhao: More on divisibility of determinants of LCM Matrices on GCD-closed sets. Southeast Asian Bull. Math. 29 (2005), 887–893. MR 2188728
[8] 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
[9] S. Hong: On the factorization of LCM matrices on gcd-closed sets. Linear Algebra Appl. 345 (2002), 225–233. MR 1883274 | Zbl 0995.15006
[10] 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
[11] S. Hong: Factorization of matrices associated with classes of arithmetical functions. Colloq. Math. 98 (2003), 113–123. DOI 10.4064/cm98-1-9 | MR 2032075 | Zbl 1047.11023
[12] S. Hong: Notes on power LCM matrices. Acta Arith. 111 (2004), 165–177. DOI 10.4064/aa111-2-5 | MR 2039420 | Zbl 1047.11022
[13] 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
[14] 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
[15] S. Hong and R. Loewy: Asymptotic behavior of eigenvalues of greatest common divisor matrices. Glasgow Math. J. 46 (2004), 551–569. DOI 10.1017/S0017089504001995 | MR 2094810
[16] S. Hong and Q. Sun: Determinants of matrices associated with incidence functions on posets. Czechoslovak Math. J. 54 (2004), 431–443. DOI 10.1023/B:CMAJ.0000042382.61841.0c | MR 2059264
[17] P. Lindqvist and K. Seip: Note on some greatest common divisor matrices. Acta Arith. 84 (1998), 149–154. DOI 10.4064/aa-84-2-149-154 | MR 1614259
[18] P. J. McCarthy: A generalization of Smith’s determinant. Canad. Math. Bull. 29 (1986), 109–113. DOI 10.4153/CMB-1986-020-1 | MR 0824893 | Zbl 0588.10005
[19] H. J. S. Smith: On the value of a certain arithmetical determinant. Proc. London Math. Soc. 7 (1875–1876), 208–212.
[20] A. Wintner: Diophantine approximations and Hilbert’s space. Amer. J. Math. 66 (1944), 564–578. DOI 10.2307/2371766 | MR 0011497 | Zbl 0061.24902
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