Research article

New results on the divisibility of power GCD and power LCM matrices

  • Received: 30 April 2022 Revised: 01 July 2022 Accepted: 08 July 2022 Published: 11 August 2022
  • MSC : 11A05, 11C20, 15B36

  • Let a,b and n be positive integers and let S be a set consisting of n distinct positive integers x1,...,xn1 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 xS, if (y<x,y|z|x and y,zS)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 1i,jn), and maxxS{|{yS: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,...,xn1 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 xS, if (y<x,y|z|x and y,zS)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 1i,jn), and maxxS{|{yS: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,...,xn1 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))1i,jn=nk=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 (xS,d|x)dS. We say that S is gcd closed if S contains (xi,xj)S for all integers i and j with 1i,jn. 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 xS, if

    (y<x,y|z|x and y,zS)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):={yS: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,  BMn(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 maxxS{|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 ab 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 maxxS{|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 maxxS{|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 maxxS{|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 1r|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 RS with |R|=r such that max(R)|min(SR), and the set SR is gcd closed.

    Clearly, any r-fold gcd-closed set must be an (r1)-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 n3, then for any gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].

    (ii). If a|b and n4, then for any (n3)-fold gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].

    (iii). Let n4 and b2. If 36b, then there exists an (n4)-fold gcd-closed set S1 with |S1|=n and maxxS1{|GS1(x)|}=2 such that (S1)(Sb1). If b0,35(mod36), then there exists an (n4)-fold gcd-closed set S2 with |S2|=n and maxxS2{|GS2(x)|}=2 such that (S2)[Sb2]. If b0,11,100(mod110), then there exists an (n4)-fold gcd-closed set S3 with |S3|=n and maxxS3{|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]=nk=1x2akαa,k, (2.1)

    where

    αa,k:=d|xkdxt,xt<xk(1ξaμ)(d), (2.2)

    where μ is the Möbius function, ξa is definedby ξa(x)=xa, and 1ξaμ is the Dirichletproduct of 1ξa and μ.

    Lemma 2.2. If S is gcd closed, then

    det(Sa)=nk=1ηa,k, (2.3)

    where

    ηa,k:=d|xkdxt,xt<xk(ξaμ)(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(ξaμ)(d)=ua,

    and

    d|u(1ξaμ)(d)=ua.

    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=xa1 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 maxxS|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 maxxS{|GS(x)|}=2. Let αa,k and ηa,k be defined as in (2.2) and (2.4), respectively. Then, for any 2kn, we have

    αa,k={1xak1xak0, if GS(xk)={xk0},1xak1xak11xak2+1xak3, if GS(xk)={xk1,xk2} and (xk1,xk2)=xk3, (2.5)

    and

    ηa,k={xakxak0, if GS(xk)={xk0},xakxak1xak2+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 maxxS|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 n3 and S being gcd closed imply that S satisfies maxxS{|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 (n3)-fold gcd-closed set S must satisfy either

    x1|x2|...|xn3|xn2|xn1|xn,

    or

    x1|x2|...|xn3|xn2 and (xn,xn1)=xn2.

    So, any (n3)-fold gcd-closed set S also satisfies maxxS{|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 (n4)-fold gcd-closed sets S1,S2 and S3 with |Si|=n and maxxSi{|GSi(x)|}=2 (1i3) 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. b2,3(mod4). Let n be an integer with n4 and S1=S(h)={x1,...,xn} with

    xk=hk1, 1kn3, xn2=2hn4, xn1=7hn4, xn=28hn4

    and h2,3(mod5). By Definition 1.2, we know that S1 is (n4)-fold gcd closed. Since GS(h)(xk)={hk2} for all integers k with 2kn3, GS(h)(xn2)={hn4}, GS(h)(xn1)={hn4}, GS(h)(xn)={2hn4,7hn4}, and (2hn4,7hn4)=hn4, by Lemmas 2.2, 2.4 and (2.6), one has

    det(S(h)b)=(2b1)(7b1)(28b2b7b+1)hb(n4)(n+1)2(hb1)n4.

    So,

    det(S(h))=23×3×5h(n4)(n+1)2(h1)n4.

    We claim that 5det(S(h)b).

    First, b2,3(mod4) yields 2b10(mod5), 7b12b10(mod5) and

    28b2b7b+13b22b+1322×22+120(mod5)or332×23+120(mod5).

    Also, h2,3(mod5) implies that h is a primitive root modulo 5. So h41(mod5). Thus

    hb1h2130(mod5)or h312 (or 1) 0(mod5).

    Hence, det(Sb1)det(S1)Z holds in this case.

    Case 1-2. b0,1(mod4) and b0(mod36), namely,

    b4,5,8,9,12,13,16,17,20,21,24,25,28,29,32,33(mod36).

    Let n be an integer with n4 and S1=S(l)={x1,...,xn} with

    xk=lk1, 1kn3, xn2=2ln4, xn1=13ln4, xn=52ln4

    and l2,3,10,13,14,15(mod19). By Definition 1.2, one knows that S1 is (n4)-fold gcd closed. Since GS(l)(xk)={lk2} for all integers k with 2kn3, GS(l)(xn2)={ln4}, GS(l)(xn1)={ln4}, GS(l)(xn)={2ln4,13ln4}, and (2ln4,13ln4)=ln4, by Lemmas 2.2, 2.4 and (2.6), one derives that

    det(S(l)b)=(2b1)(13b1)(52b2b13b+1)lb(n4)(n+1)2(lb1)n4.

    So,

    det(S(l))=23×3×19l(n4)(n+1)2(l1)n4.

    We assert that 19det(S(l)b).

    Since

    b4,5,8,9,12,13,16,17,20,21,24,25,28,29,32,33(mod36),

    one deduces that 2b10(mod19), 13b10(mod19), and

    52b2b13b+114b2b13b+114424134+1170(mod19)or 14525135+130(mod19)or 14828138+1180(mod19)or 14929139+120(mod19)or 14122121312+1130(mod19)or 14132131313+140(mod19)or 14162161316+130(mod19)or 14172171317+130(mod19)or 14202201320+150(mod19)or 14212211321+180(mod19)or 14242241324+190(mod19)or 14252251325+1180(mod19)or 14282281328+120(mod19)or 14292291329+1160(mod19)or 14322321332+1180(mod19)or 14332331333+1120(mod19).

    The condition l2,3,10,13,14,15(mod19) implies that l is a primitive root modulo 19. So, l181(mod19). Thus,

    lb1l4115,4,5,3,16,80(mod19)orl5112,14,2,13,9,10(mod19)orl818,5,16,15,3,40(mod19)orl91170(mod19)orl12110,60(mod19)orl1312,13,12,14,1,90(mod19)orl1614,16,3,8,15,50(mod19)orl1719,12,1,2,14,130(mod19)orl2013,8,4,16,5,150(mod19)orl2117,110(mod19)orl2416,100(mod19)
    orl25113,1,14,9,2,120(mod19)orl28116,15,8,5,4,30(mod19)orl29114,9,13,1,12,20(mod19)orl3215,3,6,4,8,160(mod19)orl33111,12,70(mod19).

    Hence, det(Sb1)det(S1)Z holds as expected in this case.

    Case 2-1 b1,2(mod4). Let n4 and S2=S(r)={x1,...,xn} with

    xk=rk1, 1kn3, xn2=2rn4, xn1=17rn4, xn=68rn4

    and r2,3(mod5). Since GS(r)(xk)={rk2} for all integers k with 2kn3, GS(r)(xn2)={rn4}, GS(r)(xn1)={rn4}, GS(r)(xn)={2rn4,17rn4}, and (2rn4,17rn4)=rn4, by Lemmas 2.1, 2.4 and (2.5), we have

    det[S(r)b]=(1)n42b×17b×68b(2b1)(17b1)(134b4b+68b)rb(n4)(n+3)2(rb1)n4.

    From Lemmas 2.2, 2.4 and (2.6), one derives that

    det(S(r)b)=(2b1)(17b1)(68b2b17b+1)rb(n4)(n+1)2(rb1)n4.

    So,

    det(S(r))=25×52r(n4)(n+1)2(r1)n4.

    One claims that 5det[S(r)b].

    First, b1,2(mod4) yields 2b10(mod5), 17b12b10(mod5) and

    68b34b4b+13b24b+132×22+110(mod5)or322×24+130(mod5).

    Also, r2,3(mod5) implies that r is a primitive root modulo 5. So, r41(mod5). Thus,

    rb1r11 (or 2)0(mod5)or r2130(mod5).

    Hence, det[Sb2]det(S2)Z holds as required in this case.

    Case 2-2. b0,3(mod4) and b0,35(mod36), namely,

    b4,7,8,11,12,15,16,19,20,23,24,27,28,31,32(mod36).

    Let n be an integer with n4 and S2=S(l)={x1,...,xn} with

    xk=lk1, 1kn3, xn2=2ln4, xn1=13ln4, xn=52ln4

    and l2,3,10,13,14,15(mod19). Since GS(l)(xk)={lk2} for all integers k with 2kn3, GS(l)(xn2)={ln4}, GS(l)(xn1)={ln4}, GS(l)(xn)={2ln4,13ln4}, and (2ln4,13ln4)=ln4, by Lemmas 2.1, 2.4 and (2.5), one has

    det[S(l)b]=(1)n42b×13b×52b(2b1)(13b1)(126b4b+52b)lb(n4)(n+3)2(lb1)n4.

    By Lemmas 2.2, 2.4 and (2.6), one has

    det(S(l)b)=(2b1)(13b1)(52b2b13b+1)lb(n4)(n+1)2(lb1)n4.

    So,

    det(S(l))=23×3×19l(n4)(n+1)2(l1)n4.

    One asserts that 19det[S(l)b].

    Since

    b4,7,8,11,12,15,16,19,20,23,24,27,28,31,32(mod36),

    we have 2b10(mod19),13b10(mod19) and

    52b26b4b+114b7b4b+11447444+120(mod19)or 1477747+1100(mod19)or 1487848+180(mod19)or 1411711411+160(mod19)or 1412712412+140(mod19)or 1415715415+110(mod19)or 1416716416+140(mod19)or 1419719419+140(mod19)or 1420720420+1180(mod19)or 1423723423+120(mod19)or 1424724424+1150(mod19)or 1427727427+1170(mod19)or 1428728428+1140(mod19)or 1431731431+160(mod19)or 1432732432+110mod19.

    The condition l2,3,10,13,14,15(mod19) means that l is a primitive root modulo 19. So, l181(mod19). Therefore,

    lb1l4115,4,5,3,16,80(mod19)orl7113,1,14,9,2,120(mod19)orl818,5,16,15,3,40(mod19)orl11114,9,13,1,12,20(mod19)orl12110,60(mod19)orl15111,70(mod19)orl1614,16,3,8,15,50(mod19)orl1911,2,9,12,13,140mod19)orl2013,8,4,16,5,150(mod19)orl23112,14,2,13,9,10(mod19)orl2416,100(mod19)orl271170(mod19)orl28116,15,8,5,4,30(mod19)orl3112,13,12,14,1,90(mod19)orl3215,3,6,4,8,160(mod19).

    Hence, det[Sb2]det(S2)Z holds as desired in this case.

    Case 3-1. b2,3,4,5,6,7,8,9(mod10). Let n be an integer with n4 and S3=S(h)={x1,...,xn} with

    xk=hk1, 1kn3, xn2=2hn4, xn1=7hn4, xn=28hn4

    and h2,6,7,8(mod11). Since GS(h)(xk)={hk2} for all integers k with 2kn3, GS(h)(xn2)={hn4},GS(h)(xn1)={hn4},GS(h)(xn)={2hn4,7hn4}, and (2hn4,7hn4)=hn4, by Lemmas 2.1, 2.4 and (2.5), it can be derived that

    det[S(h)b]=(1)n42b×7b×28b(2b1)(7b1)(114b4b+28b)hb(n4)(n+3)2(hb1)n4.

    So,

    det[S(h)]=(1)n424×3×72×11h(n4)(n+3)2(h1)n4.

    We claim that 11det[S(h)b].

    First, we have 2b10(mod11), 7b10(mod11) and

    28b14b4b+16b3b4b+1623242+110(mod11)or633343+150(mod11)or643444+130(mod11)or653545+190(mod11)or663646+1100(mod11)or673747+160(mod11)or683848+120(mod11)or693949+170(mod11),

    since b2,3,4,5,6,7,8,9(mod10). Also, h2,6,7,8(mod11) implies that h is a primitive root modulo 11. So, h101(mod11). Thus,

    hb1h213,2,4,80(mod11)or h317,6,1,50(mod11)or h414,8,2,30(mod11)or h5190(mod11)or h618,4,3,20(mod11)or h716,7,5,10(mod11)or h812,3,8,40(mod11)or h915,1,7,60(mod11).

    Hence, det[Sb3]det[S3]Z holds in this case.

    Case 3-2. b0,1(mod10) and b0,11,100(mod110), namely,

    b10,20,21,30,31,40,41,50,51,60,61,70,71,80,81,90,91,101(mod110).

    Let n4 and S3=S(l)={x1,...,xn} with

    xk=lk1, 1kn3, xn2=2ln4, xn1=13ln4, xn=52ln4

    and

    l2,3,4,5,7,9,10,11,13,14,15,16,17,18,19,20,21(mod23).

    Since GS(l)(xk)={lk2} for all integers k with 2kn3, GS(l)(xn2)={ln4}, GS(l)(xn1)={ln4}, GS(l)(xn)={2ln4,13ln4}, and (2ln4,13ln4)=ln4, from Lemmas 2.1, 2.4 and (2.5), one has

    det[S(l)b]=(1)n42b×13b×52b(2b1)(13b1)(126b4b+52b)lb(n4)(n+3)2(lb1)n4.

    So,

    det[S(l)]=(1)n425×3×132×23l(n4)(n+3)2(l1)n4.

    We assert that 23det[S(l)b].

    The condition

    b10,20,21,30,31,40,41,50,51,60,61,70,71,80,81,90,91,101(mod110)

    yields 2b10(mod110), 13b10(mod110) and

    52b26b4b+16b3b4b+1610310410+1140(mod23)or620320420+190(mod23)or621321421+1140(mod23)or630330430+140(mod23)or631331431+190(mod23)or640340440+1170(mod23)or641341441+140(mod23)or650350450+1180(mod23)or651351451+1170(mod23)or660360460+110(mod23)or661361461+1180(mod23)or670370470+1170(mod23)or671371471+110(mod23)or680380480+1110(mod23)or681381481+1170(mod23)or690390490+1120(mod23)or691391491+1110(mod23)or610131014101+1120(mod23).

    Since

    l2,3,4,5,7,9,10,11,13,14,15,16,17,18,19,20,21(mod23),

    one can derive that

    lb1l10111,7,5,8,12,17,15,20(mod23)orl2015,17,12,11,7,1,2,80(mod23)orl21111,7,5,13,9,17,6,16,15,4,19,12,10,8,140(mod23)orl3012,5,8,15,11,12,1,30(mod23)orl3115,17,12,10,14,19,2,20,13,7,11,9,4,160(mod23)orl40112,1,7,5,17,3,8,110(mod23)orl4112,5,8,6,10,12,20,19,1,9,18,11,21,15,13,160(mod23)orl50117,15,1,7,3,2,5,120(mod23)orl51112,1,7,16,4,3,13,18,8,10,17,5,14,20,90(mod23)orl6018,12,11,2,5,7,3,2,150(mod23)orl61117,15,1,14,18,2,16,9,19,3,20,7,6,40(mod23)orl70115,11,2,3,8,5,17,10(mod23)orl7118,12,11,19,16,7,18,20,3,14,6,5,9,2,10,130(mod23)orl8017,3,17,12,1,15,11,50(mod23)
    orl81115,11,2,18,13,5,4,17,16,20,8,3,19,10,60(mod23)orl9013,8,15,1,2,11,7,170(mod23)orl9117,3,17,9,20,15,10,21,11,6,16,1,19,12,4,18,140(mod23)orl10113,8,15,20,19,11,14,13,7,10,4,2,1,6,180(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 n3, then for any gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb]. Furthermore, if a|b and n4, then for any (n3)-fold gcd-closed set S with |S|=n, one has (Sa)|(Sb), (Sa)|[Sb] and [Sa]|[Sb].

    On the other hand, let n4, b2 be integers with 36b (resp. b0,35(mod36), or b0,11,100(mod110)). By part (iii) of Theorem 1.3 in this paper, we know that there exist some (n4)-fold gcd-closed sets S with maxxS{|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. b0,35(mod36), or b0,11,100(mod110)), does there exist an (n4)-fold gcd-closed set S with maxxS{|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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