Geodesic
On Hong’s conjecture for power LCM matrices
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 253-268 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $t\ge 2$.
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 $t\ge 2$.
Classification : 11A25, 11C20
Keywords: gcd-closed set; greatest-type divisor(GTD); maximal gcd-fixed set(MGFS); least common multiple matrix; power LCM matrix; nonsingularity 
@article{CMJ_2007_57_1_a20,
     author = {Cao, Wei},
     title = {On {Hong{\textquoteright}s} conjecture for power {LCM} matrices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {253--268},
     year = {2007},
     volume = {57},
     number = {1},
     mrnumber = {2309964},
     zbl = {1174.11030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a20/}
}
TY  - JOUR
AU  - Cao, Wei
TI  - On Hong’s conjecture for power LCM matrices
JO  - Czechoslovak Mathematical Journal
PY  - 2007
SP  - 253
EP  - 268
VL  - 57
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a20/
LA  - en
ID  - CMJ_2007_57_1_a20
ER  - 
%0 Journal Article
%A Cao, Wei
%T On Hong’s conjecture for power LCM matrices
%J Czechoslovak Mathematical Journal
%D 2007
%P 253-268
%V 57
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a20/
%G en
%F CMJ_2007_57_1_a20
Cao, Wei. On Hong’s conjecture for power LCM matrices. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 253-268. http://geodesic.mathdoc.fr/item/CMJ_2007_57_1_a20/

[1] S. Beslin: Reciprocal GCD matrices and LCM matrices. Fibonacci Quart. 29 (1991), 271–274. | MR | Zbl

[2] S. Beslin and S. Ligh: Greatest common divisor matrices. Linear Algebra Appl. 118 (1989), 69–76. | DOI | MR

[3] K. Bourque and S. Ligh: Matrices associated with classes of arithmetical functions. J. Number Theory 45 (1993), 367–376. | DOI | MR

[4] K. Bourque and S. Ligh: On GCD and LCM matrices. Linear Algebra Appl. 174 (1992), 65–74. | DOI | MR

[5] K. Bourque and S. Ligh: Matrices associated with classes of multiplicative functions. Linear Algebra Appl. 216 (1995), 267–275. | MR

[6] S. Z. Chun: GCD and LCM power matrices. Fibonacci Quart. 34 (1996), 290–297. | MR

[7] P. Haukkanen, J. Wang and J. Sillanpää: On Smith’s determinant. Linear Algebra Appl. 258 (1997), 251–269. | MR

[8] S. Hong: LCM matrix on an r-fold gcd-closed set. J. Sichuan Univ. Nat. Sci. Ed. 33 (1996), 650–657. | MR | Zbl

[9] S. Hong: On Bourque-Ligh conjecture of LCM matrices. Adv. in Math. (China) 25 (1996), 566–568. | MR | Zbl

[10] S. Hong: On LCM matrices on GCD-closed sets. Southeast Asian Bull. Math. 22 (1998), 381–384. | MR | Zbl

[11] S. Hong: On the Bourque-Ligh conjecture of least common multiple matrices. J. Algebra 218 (1999), 216–228. | DOI | MR | Zbl

[12] S. Hong: Gcd-closed sets and determinants of matrices associated with arithmetical functions. Acta Arith. 101 (2002), 321–332. | DOI | MR | Zbl

[13] S. Hong: On the factorization of LCM matrices on gcd-closed sets. Linear Algebra Appl. 345 (2002), 225–233. | MR | Zbl

[14] S. Hong: Notes on power LCM matrices. Acta Arith. 111 (2004), 165–177. | DOI | MR | Zbl

[15] S. Hong: Nonsingularity of matrices associated with classes of arithmetical functions. J.  Algebra 281 (2004), 1–14. | DOI | MR | Zbl

[16] S. Hong: Nonsingularity of least common multiple matrices on gcd-closed sets. J. Number Theory 113 (2005), 1–9. | DOI | MR | Zbl

[17] H. J. S. Smith: On the value of a certain arithmetical determinant. Proc. London Math. Soc. 7 (1875–1876), 2080–212.