Let a,b and n be positive integers and let S be a set consisting of n distinct positive integers x1,...,xn−1 and xn. Let (Sa) (resp. [Sa]) denote the n×n matrix having gcd(xi,xj)a (resp. lcm(xi,xj)a) as its (i,j)-entry. For any integer x∈S, if (y<x,y|z|x and y,z∈S)⇒z∈{y,x}, then y is called a greatest-type divisor of x in S. In this paper, we establish some results about the divisibility between (Sa) and (Sb), between (Sa) and [Sb] and between [Sa] and [Sb] when a|b, S is gcd closed (i.e., gcd(xi,xj)∈S for all 1≤i,j≤n), and maxx∈S{|{y∈S:y is a greatest-type divisor of x in S}|}=2.
Citation: Guangyan Zhu, Mao Li, Xiaofan Xu. New results on the divisibility of power GCD and power LCM matrices[J]. AIMS Mathematics, 2022, 7(10): 18239-18252. doi: 10.3934/math.20221003
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Let a,b and n be positive integers and let S be a set consisting of n distinct positive integers x1,...,xn−1 and xn. Let (Sa) (resp. [Sa]) denote the n×n matrix having gcd(xi,xj)a (resp. lcm(xi,xj)a) as its (i,j)-entry. For any integer x∈S, if (y<x,y|z|x and y,z∈S)⇒z∈{y,x}, then y is called a greatest-type divisor of x in S. In this paper, we establish some results about the divisibility between (Sa) and (Sb), between (Sa) and [Sb] and between [Sa] and [Sb] when a|b, S is gcd closed (i.e., gcd(xi,xj)∈S for all 1≤i,j≤n), and maxx∈S{|{y∈S:y is a greatest-type divisor of x in S}|}=2.
For arbitrary integers x and y, we denote by (x,y) (resp. [x,y]) the greatest common divisor (resp. least common multiple) of integers x and y. Let a,b and n be positive integers. Let S be a set consisting of n distinct positive integers x1,...,xn−1 and xn. Let (Sa) (resp. [Sa]) stand for the n×n matrix with (xi,xj)a (resp. [xi,xj]a) as its (i,j)-entry, which is called ath power GCD matrix (resp. ath power LCM matrix). In 1875, Smith [19] proved that
det((i,j))1≤i,j≤n=n∏k=1φ(k), | (1.1) |
where φ is the Euler's phi function. After that, many generalizations of Smith's determinant (1.1) were published (see, for instance, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and [20,21,22,23,24,25,26,27]).
The set S is said to be factor closed (FC) if (x∈S,d|x)⇒d∈S. We say that S is gcd closed if S contains (xi,xj)∈S for all integers i and j with 1≤i,j≤n. Obviously, an FC set must be gcd closed, but the converse is not true. As usual, let Z and |S| denote the ring of integers and the cardinality of the set S, respectively. In 1999, Hong [10] introduced the concept of a greatest-type divisor when he [10] solved completely the renowned Bourque-Ligh conjecture [2]. For any integer x∈S, if
(y<x,y|z|x and y,z∈S)⇒z∈{y,x}, |
then y is called a greatest-type divisor of x. One defines a subset GS(x) of S as follows:
GS(x):={y∈S:y is a greatest-type divisor of x in S}. |
Let Mn(Z) stand for the ring of n×n matrices over the integers. Bourque and Ligh [2] proved that (S) divides [S] in the ring Mn(Z) if S is FC. Namely, ∃ B∈Mn(Z) such that [S]=B(S) or [S]=(S)B. Hong [12] showed that such a factorization is no longer true in general when S is gcd closed and maxx∈S{|GS(x)|}=2. The results of Bourque-Ligh and Hong are extended by Korkee and Haukkanen [17] and by Chen, Hong and Zhao [5]. Feng, Hong and Zhao [6], Zhao [22], Altinisik, Yildiz and Keskin [1] and Zhao, Chen and Hong [23] used the greatest-type divisor to make important progress on an open problem of Hong raised in [12].
On the other hand, Hong [15] initially studied the divisibility among power GCD matrices and among power LCM matrices. It was proved in [15] that (Sa)|(Sb),(Sa)|[Sb] and [Sa]|[Sb] if a|b and S is a divisor chain (that is, xσ(1)|...|xσ(n) for a permutation σ of {1,...,n}), and such factorizations are no longer true if a∤b and |S|≥2. Evidently, a divisor chain is gcd closed but not conversely. Recently, Zhu [24] and Zhu and Li [27] confirmed three conjectures of Hong raised in [15] stating that if a|b and S is a gcd-closed set with maxx∈S{|GS(x)|}=1, then the bth power GCD matrix (Sb) (resp. the bth power LCM matrix [Sb]) is divisible by the ath power GCD matrix (Sa), and the bth power LCM matrix [Sb] is divisible by the ath power LCM matrix (Sa). One naturally asks the following question: If a|b and S is gcd closed and maxx∈S{|GS(x)|}=2, then is it true that (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb] hold in Mn(Z)? In particular, the following interesting question arises.
Problem 1.1. Let S be a gcd-closed set with maxx∈S{|GS(x)|}=2. Is it true that (S)|(Sb), (S)|[Sb] and [S]|[Sb] hold in Mn(Z)?
In this paper, our main goal is to study Problem 1.1. To state our main result, we need the following concept, also due to Hong.
Definition 1.2. ([9,14]) Let S be a finite set of distinct positive integers, and let r be an integer with 1≤r≤|S|−1. The set S is called 0-fold gcd closed if S is gcd closed. The set S is called r-fold gcd closed if there is a divisor chain R⊂S with |R|=r such that max(R)|min(S∖R), and the set S∖R is gcd closed.
Clearly, any r-fold gcd-closed set must be an (r−1)-fold gcd-closed set, and the converse is not true. We can now state the main result of this paper.
Theorem 1.3 Let a,b and n be positive integers. Then, each of the following is true:
(i). If a|b and n≤3, then for any gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].
(ii). If a|b and n≥4, then for any (n−3)-fold gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].
(iii). Let n≥4 and b≥2. If 36∤b, then there exists an (n−4)-fold gcd-closed set S1 with |S1|=n and maxx∈S1{|GS1(x)|}=2 such that (S1)∤(Sb1). If b≢0,35(mod36), then there exists an (n−4)-fold gcd-closed set S2 with |S2|=n and maxx∈S2{|GS2(x)|}=2 such that (S2)∤[Sb2]. If b≢0,11,100(mod110), then there exists an (n−4)-fold gcd-closed set S3 with |S3|=n and maxx∈S3{|GS3(x)|}=2 such that [S3]∤[Sb3].
It is obvious that
(Sa)|(Sb)⟺(Saσ)|(Sbσ),(Sa)|[Sb]⟺(Saσ)|[Sbσ], and [Sa]|[Sb]⟺[Saσ]|[Sbσ] |
for any permutation σ on the set {1,...,n}, where Sσ:={xσ(1),...,xσ(n)}. Thus, without loss of generality, we may let x1<⋯<xn in what follows.
This paper is organized as follows. In Section 2, we supply some preliminary results that are needed in the proof of Theorem 1.3. Then, in Section 3, we present the proof of Theorem 1.3.
At first, for any arithmetic function f, we define the reciprocal arithmetic function 1f for any positive integer m by
1f(m):={0if f(m)=0,1f(m)otherwise. |
We need two known results which give the formulas for the determinants of the power LCM matrix and power GCD matrix on gcd-closed sets.
Lemma 2.1. [13,Lemma 2.1] If S is gcd closed, then
det[Sa]=n∏k=1x2akαa,k, | (2.1) |
where
αa,k:=∑d|xkd∤xt,xt<xk(1ξa∗μ)(d), | (2.2) |
where μ is the Möbius function, ξa is definedby ξa(x)=xa, and 1ξa∗μ is the Dirichletproduct of 1ξa and μ.
Lemma 2.2. If S is gcd closed, then
det(Sa)=n∏k=1ηa,k, | (2.3) |
where
ηa,k:=∑d|xkd∤xt,xt<xk(ξa∗μ)(d). | (2.4) |
Proof. This follows immediately from [3,Theorem 2] applied to f=ξa.
Lemma 2.3. Let u be a positive integer. Then,
∑d|u(ξa∗μ)(d)=ua, |
and
∑d|u(1ξa∗μ)(d)=u−a. |
Proof. The results follow immediately from [11,Lemma 7] applied to f=ξa and f=1ξa respectively.
Lemma 2.4. Let αa,k and ηa,k be definedas in (2.2) and(2.4), respectively. Then, αa,1=x−a1 and ηa,1=xa1.
Proof. Lemma 2.4 follows immediately from Lemma 2.3.
Lemma 2.5. [24, Theorem 1.3] Let S be gcd closed and maxx∈S|GS(x)|=1 andlet a and b be positive integers with a|b. Then, in the ring Mn(Z), we have (Sa)|(Sb) and (Sa)|[Sb].
Lemma 2.6. [6, Lemma 2.2] Let S be a gcd-closed set with maxx∈S{|GS(x)|}=2. Let αa,k and ηa,k be defined as in (2.2) and (2.4), respectively. Then, for any 2≤k≤n, we have
αa,k={1xak−1xak0, if GS(xk)={xk0},1xak−1xak1−1xak2+1xak3, if GS(xk)={xk1,xk2} and (xk1,xk2)=xk3, | (2.5) |
and
ηa,k={xak−xak0, if GS(xk)={xk0},xak−xak1−xak2+xak3, if GS(xk)={xk1,xk2} and (xk1,xk2)=xk3. | (2.6) |
Lemma 2.7. [27,Theorem 1.1] Let S be gcd closed and maxx∈S|GS(x)|=1 andlet a and b be positive integers with a|b. Then, in the ring Mn(Z), one has [Sa]|[Sb].
In this section, we use the lemmas presented in the previous section to give the proof of Theorem 1.3.
Proof of Theorem 1.3. First, we prove part (i). The conditions n≤3 and S being gcd closed imply that S satisfies maxx∈S{|GS(x)|}=1. It then follows immediately from Lemmas 2.5 and 2.7 that part (i) is true. Part (i) is proved.
Subsequently, we prove part (ii). First of all, any (n−3)-fold gcd-closed set S must satisfy either
x1|x2|...|xn−3|xn−2|xn−1|xn, |
or
x1|x2|...|xn−3|xn−2 and (xn,xn−1)=xn−2. |
So, any (n−3)-fold gcd-closed set S also satisfies maxx∈S{|GS(x)|}=1. Part (ii) follows immediately from Lemmas 2.5 and 2.7. Part (ii) is proved.
Finally, we show part (iii). To do so, it is sufficient to prove that there exist (n−4)-fold gcd-closed sets S1,S2 and S3 with |Si|=n and maxx∈Si{|GSi(x)|}=2 (1≤i≤3) such that
det(S1)∤det(Sb1),det(S2)∤det[Sb2],det[S3]∤det[Sb3]. |
Let us continue the proof of part (iii) of Theorem 1.3, which is divided into the following cases.
Case 1-1. b≡2,3(mod4). Let n be an integer with n≥4 and S1=S(h)={x1,...,xn} with
xk=hk−1, 1≤k≤n−3, xn−2=2hn−4, xn−1=7hn−4, xn=28hn−4 |
and h≡2,3(mod5). By Definition 1.2, we know that S1 is (n−4)-fold gcd closed. Since GS(h)(xk)={hk−2} for all integers k with 2≤k≤n−3, GS(h)(xn−2)={hn−4}, GS(h)(xn−1)={hn−4}, GS(h)(xn)={2hn−4,7hn−4}, and (2hn−4,7hn−4)=hn−4, by Lemmas 2.2, 2.4 and (2.6), one has
det(S(h)b)=(2b−1)(7b−1)(28b−2b−7b+1)hb(n−4)(n+1)2(hb−1)n−4. |
So,
det(S(h))=23×3×5h(n−4)(n+1)2(h−1)n−4. |
We claim that 5∤det(S(h)b).
First, b≡2,3(mod4) yields 2b−1≢0(mod5), 7b−1≡2b−1≢0(mod5) and
28b−2b−7b+1≡3b−2⋅2b+1≡32−2×22+1≡2≢0(mod5)or≡33−2×23+1≡2≢0(mod5). |
Also, h≡2,3(mod5) implies that h is a primitive root modulo 5. So h4≡1(mod5). Thus
hb−1≡h2−1≡3≢0(mod5)or ≡h3−1≡2 (or 1) ≢0(mod5). |
Hence, det(Sb1)det(S1)∉Z holds in this case.
Case 1-2. b≡0,1(mod4) and b≢0(mod36), namely,
b≡4,5,8,9,12,13,16,17,20,21,24,25,28,29,32,33(mod36). |
Let n be an integer with n≥4 and S1=S(l)={x1,...,xn} with
xk=lk−1, 1≤k≤n−3, xn−2=2ln−4, xn−1=13ln−4, xn=52ln−4 |
and l≡2,3,10,13,14,15(mod19). By Definition 1.2, one knows that S1 is (n−4)-fold gcd closed. Since GS(l)(xk)={lk−2} for all integers k with 2≤k≤n−3, GS(l)(xn−2)={ln−4}, GS(l)(xn−1)={ln−4}, GS(l)(xn)={2ln−4,13ln−4}, and (2ln−4,13ln−4)=ln−4, by Lemmas 2.2, 2.4 and (2.6), one derives that
det(S(l)b)=(2b−1)(13b−1)(52b−2b−13b+1)lb(n−4)(n+1)2(lb−1)n−4. |
So,
det(S(l))=23×3×19l(n−4)(n+1)2(l−1)n−4. |
We assert that 19∤det(S(l)b).
Since
b≡4,5,8,9,12,13,16,17,20,21,24,25,28,29,32,33(mod36), |
one deduces that 2b−1≢0(mod19), 13b−1≢0(mod19), and
52b−2b−13b+1≡14b−2b−13b+1≡144−24−134+1≡17≢0(mod19)or ≡145−25−135+1≡3≢0(mod19)or ≡148−28−138+1≡18≢0(mod19)or ≡149−29−139+1≡2≢0(mod19)or ≡1412−212−1312+1≡13≢0(mod19)or ≡1413−213−1313+1≡4≢0(mod19)or ≡1416−216−1316+1≡3≢0(mod19)or ≡1417−217−1317+1≡3≢0(mod19)or ≡1420−220−1320+1≡5≢0(mod19)or ≡1421−221−1321+1≡8≢0(mod19)or ≡1424−224−1324+1≡9≢0(mod19)or ≡1425−225−1325+1≡18≢0(mod19)or ≡1428−228−1328+1≡2≢0(mod19)or ≡1429−229−1329+1≡16≢0(mod19)or ≡1432−232−1332+1≡18≢0(mod19)or ≡1433−233−1333+1≡12≢0(mod19). |
The condition l≡2,3,10,13,14,15(mod19) implies that l is a primitive root modulo 19. So, l18≡1(mod19). Thus,
lb−1≡l4−1≡15,4,5,3,16,8≢0(mod19)or≡l5−1≡12,14,2,13,9,1≢0(mod19)or≡l8−1≡8,5,16,15,3,4≢0(mod19)or≡l9−1≡17≢0(mod19)or≡l12−1≡10,6≢0(mod19)or≡l13−1≡2,13,12,14,1,9≢0(mod19)or≡l16−1≡4,16,3,8,15,5≢0(mod19)or≡l17−1≡9,12,1,2,14,13≢0(mod19)or≡l20−1≡3,8,4,16,5,15≢0(mod19)or≡l21−1≡7,11≢0(mod19)or≡l24−1≡6,10≢0(mod19) |
or≡l25−1≡13,1,14,9,2,12≢0(mod19)or≡l28−1≡16,15,8,5,4,3≢0(mod19)or≡l29−1≡14,9,13,1,12,2≢0(mod19)or≡l32−1≡5,3,6,4,8,16≢0(mod19)or≡l33−1≡11,12,7≢0(mod19). |
Hence, det(Sb1)det(S1)∉Z holds as expected in this case.
Case 2-1 b≡1,2(mod4). Let n≥4 and S2=S(r)={x1,...,xn} with
xk=rk−1, 1≤k≤n−3, xn−2=2rn−4, xn−1=17rn−4, xn=68rn−4 |
and r≡2,3(mod5). Since GS(r)(xk)={rk−2} for all integers k with 2≤k≤n−3, GS(r)(xn−2)={rn−4}, GS(r)(xn−1)={rn−4}, GS(r)(xn)={2rn−4,17rn−4}, and (2rn−4,17rn−4)=rn−4, by Lemmas 2.1, 2.4 and (2.5), we have
det[S(r)b]=(−1)n−42b×17b×68b(2b−1)(17b−1)(1−34b−4b+68b)rb(n−4)(n+3)2(rb−1)n−4. |
From Lemmas 2.2, 2.4 and (2.6), one derives that
det(S(r)b)=(2b−1)(17b−1)(68b−2b−17b+1)rb(n−4)(n+1)2(rb−1)n−4. |
So,
det(S(r))=25×52r(n−4)(n+1)2(r−1)n−4. |
One claims that 5∤det[S(r)b].
First, b≡1,2(mod4) yields 2b−1≢0(mod5), 17b−1≡2b−1≢0(mod5) and
68b−34b−4b+1≡3b−2⋅4b+1≡3−2×22+1≡1≢0(mod5)or≡32−2×24+1≡3≢0(mod5). |
Also, r≡2,3(mod5) implies that r is a primitive root modulo 5. So, r4≡1(mod5). Thus,
rb−1≡r−1≡1 (or 2)≢0(mod5)or ≡r2−1≡3≢0(mod5). |
Hence, det[Sb2]det(S2)∉Z holds as required in this case.
Case 2-2. b≡0,3(mod4) and b≢0,35(mod36), namely,
b≡4,7,8,11,12,15,16,19,20,23,24,27,28,31,32(mod36). |
Let n be an integer with n≥4 and S2=S(l)={x1,...,xn} with
xk=lk−1, 1≤k≤n−3, xn−2=2ln−4, xn−1=13ln−4, xn=52ln−4 |
and l≡2,3,10,13,14,15(mod19). Since GS(l)(xk)={lk−2} for all integers k with 2≤k≤n−3, GS(l)(xn−2)={ln−4}, GS(l)(xn−1)={ln−4}, GS(l)(xn)={2ln−4,13ln−4}, and (2ln−4,13ln−4)=ln−4, by Lemmas 2.1, 2.4 and (2.5), one has
det[S(l)b]=(−1)n−42b×13b×52b(2b−1)(13b−1)(1−26b−4b+52b)lb(n−4)(n+3)2(lb−1)n−4. |
By Lemmas 2.2, 2.4 and (2.6), one has
det(S(l)b)=(2b−1)(13b−1)(52b−2b−13b+1)lb(n−4)(n+1)2(lb−1)n−4. |
So,
det(S(l))=23×3×19l(n−4)(n+1)2(l−1)n−4. |
One asserts that 19∤det[S(l)b].
Since
b≡4,7,8,11,12,15,16,19,20,23,24,27,28,31,32(mod36), |
we have 2b−1≢0(mod19),13b−1≢0(mod19) and
52b−26b−4b+1≡14b−7b−4b+1≡144−74−44+1≡2≢0(mod19)or ≡147−77−47+1≡10≢0(mod19)or ≡148−78−48+1≡8≢0(mod19)or ≡1411−711−411+1≡6≢0(mod19)or ≡1412−712−412+1≡4≢0(mod19)or ≡1415−715−415+1≡1≢0(mod19)or ≡1416−716−416+1≡4≢0(mod19)or ≡1419−719−419+1≡4≢0(mod19)or ≡1420−720−420+1≡18≢0(mod19)or ≡1423−723−423+1≡2≢0(mod19)or ≡1424−724−424+1≡15≢0(mod19)or ≡1427−727−427+1≡17≢0(mod19)or ≡1428−728−428+1≡14≢0(mod19)or ≡1431−731−431+1≡6≢0(mod19)or ≡1432−732−432+1≡1≢0mod19. |
The condition l≡2,3,10,13,14,15(mod19) means that l is a primitive root modulo 19. So, l18≡1(mod19). Therefore,
lb−1≡l4−1≡15,4,5,3,16,8≢0(mod19)or≡l7−1≡13,1,14,9,2,12≢0(mod19)or≡l8−1≡8,5,16,15,3,4≢0(mod19)or≡l11−1≡14,9,13,1,12,2≢0(mod19)or≡l12−1≡10,6≢0(mod19)or≡l15−1≡11,7≢0(mod19)or≡l16−1≡4,16,3,8,15,5≢0(mod19)or≡l19−1≡1,2,9,12,13,14≢0mod19)or≡l20−1≡3,8,4,16,5,15≢0(mod19)or≡l23−1≡12,14,2,13,9,1≢0(mod19)or≡l24−1≡6,10≢0(mod19)or≡l27−1≡17≢0(mod19)or≡l28−1≡16,15,8,5,4,3≢0(mod19)or≡l31−1≡2,13,12,14,1,9≢0(mod19)or≡l32−1≡5,3,6,4,8,16≢0(mod19). |
Hence, det[Sb2]det(S2)∉Z holds as desired in this case.
Case 3-1. b≡2,3,4,5,6,7,8,9(mod10). Let n be an integer with n≥4 and S3=S(h)={x1,...,xn} with
xk=hk−1, 1≤k≤n−3, xn−2=2hn−4, xn−1=7hn−4, xn=28hn−4 |
and h≡2,6,7,8(mod11). Since GS(h)(xk)={hk−2} for all integers k with 2≤k≤n−3, GS(h)(xn−2)={hn−4},GS(h)(xn−1)={hn−4},GS(h)(xn)={2hn−4,7hn−4}, and (2hn−4,7hn−4)=hn−4, by Lemmas 2.1, 2.4 and (2.5), it can be derived that
det[S(h)b]=(−1)n−42b×7b×28b(2b−1)(7b−1)(1−14b−4b+28b)hb(n−4)(n+3)2(hb−1)n−4. |
So,
det[S(h)]=(−1)n−424×3×72×11h(n−4)(n+3)2(h−1)n−4. |
We claim that 11∤det[S(h)b].
First, we have 2b−1≢0(mod11), 7b−1≢0(mod11) and
28b−14b−4b+1≡6b−3b−4b+1≡62−32−42+1≡1≢0(mod11)or≡63−33−43+1≡5≢0(mod11)or≡64−34−44+1≡3≢0(mod11)or≡65−35−45+1≡9≢0(mod11)or≡66−36−46+1≡10≢0(mod11)or≡67−37−47+1≡6≢0(mod11)or≡68−38−48+1≡2≢0(mod11)or≡69−39−49+1≡7≢0(mod11), |
since b≡2,3,4,5,6,7,8,9(mod10). Also, h≡2,6,7,8(mod11) implies that h is a primitive root modulo 11. So, h10≡1(mod11). Thus,
hb−1≡h2−1≡3,2,4,8≢0(mod11)or ≡h3−1≡7,6,1,5≢0(mod11)or ≡h4−1≡4,8,2,3≢0(mod11)or ≡h5−1≡9≢0(mod11)or ≡h6−1≡8,4,3,2≢0(mod11)or ≡h7−1≡6,7,5,1≢0(mod11)or ≡h8−1≡2,3,8,4≢0(mod11)or ≡h9−1≡5,1,7,6≢0(mod11). |
Hence, det[Sb3]det[S3]∉Z holds in this case.
Case 3-2. b≡0,1(mod10) and b≢0,11,100(mod110), namely,
b≡10,20,21,30,31,40,41,50,51,60,61,70,71,80,81,90,91,101(mod110). |
Let n≥4 and S3=S(l)={x1,...,xn} with
xk=lk−1, 1≤k≤n−3, xn−2=2ln−4, xn−1=13ln−4, xn=52ln−4 |
and
l≡2,3,4,5,7,9,10,11,13,14,15,16,17,18,19,20,21(mod23). |
Since GS(l)(xk)={lk−2} for all integers k with 2≤k≤n−3, GS(l)(xn−2)={ln−4}, GS(l)(xn−1)={ln−4}, GS(l)(xn)={2ln−4,13ln−4}, and (2ln−4,13ln−4)=ln−4, from Lemmas 2.1, 2.4 and (2.5), one has
det[S(l)b]=(−1)n−42b×13b×52b(2b−1)(13b−1)(1−26b−4b+52b)lb(n−4)(n+3)2(lb−1)n−4. |
So,
det[S(l)]=(−1)n−425×3×132×23l(n−4)(n+3)2(l−1)n−4. |
We assert that 23∤det[S(l)b].
The condition
b≡10,20,21,30,31,40,41,50,51,60,61,70,71,80,81,90,91,101(mod110) |
yields 2b−1≢0(mod110), 13b−1≢0(mod110) and
52b−26b−4b+1≡6b−3b−4b+1≡610−310−410+1≡14≢0(mod23)or≡620−320−420+1≡9≢0(mod23)or≡621−321−421+1≡14≢0(mod23)or≡630−330−430+1≡4≢0(mod23)or≡631−331−431+1≡9≢0(mod23)or≡640−340−440+1≡17≢0(mod23)or≡641−341−441+1≡4≢0(mod23)or≡650−350−450+1≡18≢0(mod23)or≡651−351−451+1≡17≢0(mod23)or≡660−360−460+1≡1≢0(mod23)or≡661−361−461+1≡18≢0(mod23)or≡670−370−470+1≡17≢0(mod23)or≡671−371−471+1≡1≢0(mod23)or≡680−380−480+1≡11≢0(mod23)or≡681−381−481+1≡17≢0(mod23)or≡690−390−490+1≡12≢0(mod23)or≡691−391−491+1≡11≢0(mod23)or≡6101−3101−4101+1≡12≢0(mod23). |
Since
l≡2,3,4,5,7,9,10,11,13,14,15,16,17,18,19,20,21(mod23), |
one can derive that
lb−1≡l10−1≡11,7,5,8,12,17,15,2≢0(mod23)or≡l20−1≡5,17,12,11,7,1,2,8≢0(mod23)or≡l21−1≡11,7,5,13,9,17,6,16,15,4,19,12,10,8,14≢0(mod23)or≡l30−1≡2,5,8,15,11,12,1,3≢0(mod23)or≡l31−1≡5,17,12,10,14,19,2,20,13,7,11,9,4,16≢0(mod23)or≡l40−1≡12,1,7,5,17,3,8,11≢0(mod23)or≡l41−1≡2,5,8,6,10,12,20,19,1,9,18,11,21,15,13,16≢0(mod23)or≡l50−1≡17,15,1,7,3,2,5,12≢0(mod23)or≡l51−1≡12,1,7,16,4,3,13,18,8,10,17,5,14,20,9≢0(mod23)or≡l60−1≡8,12,11,2,5,7,3,2,15≢0(mod23)or≡l61−1≡17,15,1,14,18,2,16,9,19,3,20,7,6,4≢0(mod23)or≡l70−1≡15,11,2,3,8,5,17,1≢0(mod23)or≡l71−1≡8,12,11,19,16,7,18,20,3,14,6,5,9,2,10,13≢0(mod23)or≡l80−1≡7,3,17,12,1,15,11,5≢0(mod23) |
or≡l81−1≡15,11,2,18,13,5,4,17,16,20,8,3,19,10,6≢0(mod23)or≡l90−1≡3,8,15,1,2,11,7,17≢0(mod23)or≡l91−1≡7,3,17,9,20,15,10,21,11,6,16,1,19,12,4,18,14≢0(mod23)or≡l101−1≡3,8,15,20,19,11,14,13,7,10,4,2,1,6,18≢0(mod23). |
So, det[Sb3]det[S3]∉Z holds as one expects in this case.
This finishes the proof of Theorem 1.3.
Let a,b and n be positive integers. Parts (i) and (ii) of Theorem 1.3 in this paper tell us that if a|b and n≤3, then for any gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb]. Furthermore, if a|b and n≥4, then for any (n−3)-fold gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].
On the other hand, let n≥4, b≥2 be integers with 36∤b (resp. b≢0,35(mod36), or b≢0,11,100(mod110)). By part (iii) of Theorem 1.3 in this paper, we know that there exist some (n−4)-fold gcd-closed sets S with maxx∈S{|GS(x)|}=2 such that in the ring M|S|(Z), one has (S)∤(Sb) (resp. (S)∤[Sb], or [S]∤[Sb]). However, when 36|b (resp. b≡0,35(mod36), or b≡0,11,100(mod110)), does there exist an (n−4)-fold gcd-closed set S with maxx∈S{|GS(x)|}=2 such that in the ring M|S|(Z), we have (S)∤(Sb) (resp. (S)∤[Sb], or [S]∤[Sb])? This question remains open.
The authors would like to thank the anonymous referees for careful reading of the manuscript and helpful comments. The corresponding author X.F. Xu was supported partially by the Foundation of Sichuan Tourism University under Grant # 20SCTUTY01 and also supported by the Initial Scientific Research Fund of Doctors in Sichuan Tourism University.
We declare that we have no conflict of interest.
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