Share at least half the numbers in a nontrivial LCM-closed set a nontrivial divisor?
aa r X i v : . [ m a t h . N T ] A ug Share at least half the numbers in a nontrivialLCM-closed set a nontrivial divisor?
Tom Fischer ∗ University of WuerzburgAugust 29, 2018
Abstract
For a finite set of non-zero natural numbers that contains at least one elementdifferent from 1 and the least common multiple of any of its subsets, there existsa subset of at least half of its members which has a common divisor larger than1. Utilizing a representation of the natural numbers as an order-theoretical ringof prime power sets, this conjecture is shown to be equivalent to Frankl’s union-closed sets conjecture. Some results for cases where the conjecture, which alsohas meaningful interpretations in graph and lattice theory, is known to hold areprovided. An equivalent dual version of the conjecture is, that for a finite setof non-zero natural numbers that contains at least two elements and the greatestcommon divisor of any of its subsets, one of its members has a prime power that isnot a prime power of more than half of the members.
Key words:
GCD-closed, abundant divisor, LCM-closed, least common multiple,union-closed sets conjecture.
MSC2010:
For i ∈ N + = N \ { } , let p i denote the i -th prime number. Ubiquitously known asthe Fundamental Theorem of Arithmetic (see [2] for an early reference), there exists auniquely determined injective function q : N + −→ N N + (1) n ( q ( n ) , q ( n ) , . . . )such that n = Y i ∈ N + ( p i ) q i ( n ) . (2) ∗ Institute of Mathematics, University of Wuerzburg, Emil-Fischer-Strasse 30, 97074 Wuerzburg, Ger-many. Phone: +49 931 3188911. E-mail: [email protected] . n . With the exception of q (1) = (0 , , . . . ), the members of q ( N + ) are the finite sequencesof N N + in the sense that any element of q ( N + ) is constant zero from certain member ofthe sequence onwards. Thus, q is a by (2) uniquely determined bijection between thenon-zero natural numbers and the finite sequences of elements of N in the thus explainedsense, including the sequence of zeros only.Also well known is that the least common multiple (LCM) of m numbers n j ∈ N + , j = 1 , . . . , m , where m ∈ N + \ { } , is given bylcm( n , . . . , n m ) = Y i ∈ N + ( p i ) max { q i ( n ) ,...,q i ( n m ) } , (3)and their greatest common divisor (GCD) bygcd( n , . . . , n m ) = Y i ∈ N + ( p i ) min { q i ( n ) ,...,q i ( n m ) } . (4)For finite N ⊂ N + with N >
1, the notation lcm( N ) and gcd( N ) with the obviousfrom (3) and (4) derived meaning will be used. In the following, the subset symbol ⊂ includes equality. DEFINITION 1.
Nonempty
N ⊂ N + is LCM-closed, respectively GCD-closed, if m, n ∈N implies lcm( m, n ) ∈ N , respectively gcd( m, n ) ∈ N . By induction, it is fairly obvious that the properties LCM-closed, or GCD-closed,apply to finite subsets of an LCM-closed, respectively GCD-closed, set N in the sensethat N ⊂ N + is LCM-closed, respectively GCD-closed, if and only if M ⊂ N implieslcm( M ) ∈ N , respectively gcd( M ) ∈ N , for any finite M . EXAMPLE 1. N = { , , , , , } is LCM- and GCD-closed.2. N = { , , , , } is LCM-closed, but not GCD-closed.3. N = { , , , , , , } is GCD-closed, but not LCM-closed. EXAMPLE 2.
Consider a dynamical system on a nonempty set X given by a map s : X → X . A nonempty subset A ⊂ X is periodic if P ( A ) = { n ∈ N + \ { } : s n ( x ) = x for all x ∈ A } 6 = ∅ , and P f ( A ) := min P ( A ) is then called the fundamental period of A .Given P ( A ) = ∅ , the set { P f ( B ) : B ⊂ A } is LCM-closed, and P f ( A ) is its maximum. Ifnonempty, { P f ( A ) : A ⊂ X } is LCM-closed, but potentially infinite. CONJECTURE 1.
For an LCM-closed finite set of non-zero natural numbers that con-tains at least one element different from 1, there exists a divisor larger than 1 for a subsetof at least half of its members.
EXAMPLE 3.
The set N = { , , , , , , , } of eight elements is LCM-closed. Thesubset { , , , , , } of six elements has the greatest common divisor 2.The next example shows that an abundant divisor in an LCM-closed set neither hasto be a member of the set, nor does it have to be prime. EXAMPLE 4.
The set N = { , , , , , , } is LCM-closed, but not GCD-closed. The numbers 2, 3, 5, 6, 7, 10, and 14 are abundant nontrivial divisors.The following set illustrates that not each prime factor in lcm( N ) must be an abundantdivisor in an LCM-closed set N . EXAMPLE 5.
The set N = { , , } is LCM-closed, but 5 only divides element 30.The main result of this note will be the equivalence of Conjecture 1 to the union-closedsets conjecture, which – according to [1] and despite of a plethora of articles relating toit – has resisted proof since at least 1979. While connections to graph theory (e.g. [3])and lattice theory (e.g. [3] and [4]) are well known (see [1] for summaries), no relations tonumber theory seem to have been drawn in the past. DEFINITION 2.
A family (“system”) S of sets is union-closed, respectivelyintersection-closed, if A, B ∈ S implies A ∪ B ∈ S , respectively A ∩ B ∈ S .An order-theoretical ring is a family of sets, which is simultaneously union- andintersection-closed. Similar to Definition 1, a family S of sets is union-closed, respectively intersection-closed, if and only if it holds for finite ˜ S ⊂ S that S ˜ S ∈ S , respectively T ˜ S ∈ S .Obviously, any finite order-theoretical ring is a complete lattice (by the order of inclusion)with intersection as the meet operation and union as the join operation.
Union-closed sets conjecture.
For a finite union-closed family of sets with at leastone nonempty member, there exists an element shared by at least half of the member sets.
Bruhn and Schaudt [1] is a fairly recent and very thorough overview article on theFrankl conjecture, which is why no comprehensive summary of conditions, under whichthe conjecture is known to hold, needs to be provided here. A few selected conditions willbe given below. However, besides a simple example further down, a few basic tools andfacts regarding the conjecture should be pointed out to readers who are unfamiliar withit. Considering a union-closed family of sets, it makes sense to identify elements that liein exactly the same sets, thus considering them as one single element. Since in any hereconsidered union-closed system S the maximal set (using the order of inclusion; this set3s also called the “universe”) is finite, elements of member sets can be identified withnatural numbers. For this, let the maximal set be S max = { , , . . . , m } , m ∈ N + . Now,all members of S are subsets of natural numbers in S max , where the empty set can be amember, as well. It is easy to show – and widely known – that union-closed systems thateither have a singleton member set, or a member set with only two elements, adhere tothe conjecture. EXAMPLE 6.
The following is a union-closed system with eight member sets: S = {∅ , { } , { , } , { , , } , { } , { , } , { , , } , { , , , }} . (5)The elements 1, 2, and 4 are abundant. The universe is S max = { , , , } . Note that, forinstance, the member sets ∅ , { } , { , } , { , , } , and { } could each, or all, be removed,while the system would remain union-closed.A few cases where the conjecture is known to hold are (see [1] and references therein):1. S has a member with only one or with only two elements.2. S max ≤ S ≤
S ≥ S max .5. S ≤ S max ), if S is such that for any x, y ∈ S max there exist A, B ∈ S such that x ∈ A and y ∈ B , but x, y / ∈ A ∩ B . The system S is then called separating. N + as a ring of prime power sets In preparation for the proof of the main result, which establishes equivalence of Conjecture1 and the union-closed sets conjecture, this section examines how the natural numberscan be considered as an order-theoretical ring with regard to the operations “lcm” and“gcd”. Considerations similar to the ones carried out here, but regarding natural numbersas multisets of their prime factors, may be familiar to the reader and may indeed beconsidered as folklore knowledge. This remark extends to Lemma 1 below. However, it isbetter for the purpose of this note to not use multisets of prime factors, and instead use(proper) sets of prime powers, which – as a slightly different setup – justifies the moredetailed explanations provided here.With P ( N + ) the power set of N + , define f : N + −→ P ( N + )(6) n [ i ∈ N + { p i , ( p i ) , . . . , ( p i ) q i ( n ) } , with the convention { p i , ( p i ) , . . . , ( p i ) q i ( n ) } = ∅ for q i ( n ) = 0. The function f , where f (1) = ∅ , maps any natural number n > n , all powers of this prime below the highest one are included intothe set as well. 4 XAMPLE 7. f (18) = f (2 · ) = { , , } , but { , } , { , , } / ∈ f ( N + ). f (16) = f (2 ) = { , , , } , however, for instance, { } , { , } , { , } , { , } , { , , } , { , , } / ∈ f ( N + ).Clearly, f is an injection with the inverse g := f − : f ( N + ) −→ N + , (7)which maps finite sets of prime powers (where prime powers below the maximal oneare included) to their product, i.e. to the natural number with the corresponding primefactorization. More precisely, g maps any member set of f ( N + ) to the product of itsmaximal prime powers (with g ( ∅ ) = 1), and f : N + −→ f ( N + )(8)is a bijection with inverse g . EXAMPLE 8. g ( { , , } ) = g ( f (18)) = 2 · = 18 and g ( { , , , } ) = 2 = 16. LEMMA 1.
For m ∈ N + \ { } and n j ∈ N + for j = 1 , . . . , m , f (lcm( n , . . . , n m )) = [ j ∈{ ,...,m } f ( n j ) , (9) f (gcd( n , . . . , n m )) = \ j ∈{ ,...,m } f ( n j ) . (10) For m, n ∈ N + , m | n ⇔ f ( m ) ⊂ f ( n ) , (11) q = lcm( m, n ) ⇔ f ( q ) = f ( m ) ∪ f ( n ) , (12) q = gcd( m, n ) ⇔ f ( q ) = f ( m ) ∩ f ( n ) , (13) n prime ⇔ f ( n ) inclusion-minimal in f ( N + ) . (14) Proof.
Equations (9) and (10) follow in a straightforward manner from (3) and (4). Equiv-alence (11) holds since a divisor is the product of a subset of primes (prime powers) inthe factorization of the number it divides. The equivalences (12) and (13) follow directlywith (9) and (10). Statement (14) holds because of (11) and since f (1) = ∅ .Obviously, g ( f ( m ) ∪ f ( n )) = lcm( m, n ) , (15) g ( f ( m ) ∩ f ( n )) = gcd( m, n ) . (16) COROLLARY 1. N ⊂ N + is LCM-closed, respectively GCD-closed, if and only if f ( N ) is union-closed, respectively intersection-closed. . f ( N + ) is an order-theoretical ring.Proof. The first statement follows with (12) and (13). The second statement followsdirectly from the first one, since N + is LCM- and GCD-closed.The bijection (8) and its inverse g have thus established an isomorphism between thestrictly positive natural numbers, equipped with the operations “lcm” and “gcd”, and theorder-theoretical ring f ( N + ) of, essentially, the sets of prime powers of these numbers,equipped with the union and intersection operation. Certainly folklore knowledge, withthe explanation below Definition 2, any finite simultaneously LCM- and GCD-closed set N ⊂ N + is a complete lattice with “gcd” as the meet operation and “lcm” as the joinoperation. With (11), it is clear that the order is given by smaller elements dividing largerones.Note that while N + with “lcm” is a commutative monoid (or abelian monoid; theneutral element is 1, since lcm( n,
1) = 1), N + with “gcd” is no monoid, such that neither analgebraic ring structure, nor something similar (e.g. a ring without the negative elements)can be established on N + with regard to the operations given by the LCM and the GCD. THEOREM 1.
Conjecture 1 is equivalent to the union-closed sets conjecture.Proof. ⇒ : Assume that an arbitrary finite union-closed system, S , with a maximal set S max = { , , . . . , m } , m ∈ N + is given. In a first step, replace all sets of S with setsof corresponding prime numbers by means of replacing any number i ∈ { , . . . , m } withthe i -th prime number p i . For instance, one now has S max = { p , p , . . . , p m } . By thisprocedure, a set-theoretical isomorphism has been established between the two systems,which preserves operations such as unions and intersections, or relationships such asinclusion. Clearly, under this isomorphism, the system’s set-related properties such asunion-closedness, or frequencies of the occurrence of elements in member sets, remainunchanged. The elements of member sets of S are now prime numbers, so S ⊂ f ( N + )and g ( S ) ⊂ N + . Since f is a bijection with inverse g (cf. (8)), f ( g ( S )) = S . By Corollary1, g ( S ) is LCM-closed, since S is union-closed by assumption. Under Conjecture 1, asubfamily of numbers N ⊂ g ( S ) comprised of more than half the numbers in g ( S ) sharesa divisor larger than one, which implies that all numbers in N have at least one commonprime number (or even a product of primes) in their corresponding factorizations. Thus,all members of f ( N ) ⊂ S , which are at least half of the members of S , share thisprime number (or these prime numbers in the corresponding product) as an element (aselements). ⇐ : Assume that N ⊂ N + is nonempty, finite, LCM-closed, and contains at least oneelement larger than 1. By Corollary 1, S := f ( N ) is a finite, union-closed system with– by (6) – at least one nonempty member. Under the union-closed sets conjecture, asubfamily ˜ S ⊂ S = f ( N ) contains at least half of the members of S = f ( N ), and theyall share at least one element. By construction, elements of member sets S ∈ S = f ( N )are prime powers in the factorization of g ( S ) ∈ N . The restriction f : N → f ( N ) being6 bijection with inverse g , this means that at least half of the members of N share thesame prime power, and thus have a non-trivial divisor. EXAMPLE 9.
The two directions of the proof are illustrated. ⇒ : Using S of Example 6 as an instance, the union-closed system S = {∅ , { } , { , } , { , , } , { } , { , } , { , , } , { , , , }} (17)is, by i p i , first turned into the set-isomorphic S = {∅ , { } , { , } , { , , } , { } , { , } , { , , } , { , , , }} , (18)which produces the LCM-closed set (note that 210 = 2 · · · g ( S ) = { , , , , , , , } . (19) ⇐ : Using N of Example 3 as an instance, the LCM-closed set N = { , , , , , , , } (20) = { , , , , · , , · , · } is turned into the union-closed system f ( N ) = {∅ , { } , { } , { , } , { , } , { , , } , { , , } , { , , , }} . (21)The bijection 1 A well-known dual equivalent of the union-closed sets conjecture exists (e.g. [1]).
Intersection-closed sets conjecture.
For a finite intersection-closed family of sets withat least two member sets, there exists an element shared by at most half of the member sets.
CONJECTURE 2.
For a GCD-closed finite set of non-zero natural numbers that con-tains at least two elements, one of its members has a prime power that is not a primepower of more than half of the members.
A prime power as in Conjecture 2 will be called non-abundant.
EXAMPLE 10.
Revisit the GCD-closed set N = { , , , , , , } of Example 1, whichhas seven elements. The prime powers 2 , 2 , and 3 are non-abundant. COROLLARY 2.
Conjecture 2 is equivalent to the intersection-closed sets conjecture,and thus to the union-closed sets conjecture, and to Conjecture 1.Proof.
The proof of equivalence for Conjecture 2 and the intersection-closed sets conjec-ture follows in very close analogy to the proof of Theorem 1, with the obviously necessaryreplacements of “union” by “intersection”, “LCM” by “GCD”, and “Conjecture 1” by“Conjecture 2”.While Conjecture 2 has been established as the dual of Conjecture 1, it may still beof interest to look at e.g. the specific LCM-closed dual of a given GCD-closed system. Inorder to do this, define for any finite
N ∈ N + , and withˆ n := lcm( N ) , (23)a function h on N by h : n = Y i ∈ N + ( p i ) q i ( n ) Y i ∈ N + ( p i ) q i (ˆ n ) − q i ( n ) , (24)and denote N ∗ = h ( N ) . (25) EXAMPLE 11.
For the GCD-closed set N = { , , , , , , } of Example 10 (andEx. 1), lcm( N ) = 24 = 2 ·
3, such that N ∗ = h ( N ) = { , , , , , , } . For instance, h (1) = 2 − · − = 24 , (26) h (12) = h (2 ·
3) = 2 − · − = 2 . (27) 8 EMMA 2.
For any finite
N ∈ N + , it holds that N ∗ = h ( N ) ⊂ N + and h : N −→ N ∗ (28) is a bijection.Proof. For all n ∈ N , the exponents q i (ˆ n ) − q i ( n ) in (24) are non-negative because of (3).Thus, (25) is a subset of N + . Since q of (1) is an injection, h : N → N ∗ is a bijection. PROPOSITION 1.
1. Let
N ∈ N + be a GCD-closed set adhering to Conjecture 2. Then N ∗ = h ( N ) is anLCM-closed set adhering to Conjecture 1.2. Let N ∈ N + be an LCM-closed set with at least two members adhering to Conjecture1. Then N ∗ = h ( N ) is a GCD-closed set adhering to Conjecture 2.Proof.
1. By Lemma 2, N ∗ is finite and has more than one element (since it has as manyas N , which has at least two), thus containing an element larger than 1. Consider nowtwo elements m ∗ , n ∗ ∈ N ∗ with pre-images m, n ∈ N under h . By (24), q i ( m ∗ ) = q i ( h ( m )) = q i (ˆ n ) − q i ( m ) , (29) q i ( n ∗ ) = q i ( h ( n )) = q i (ˆ n ) − q i ( n ) . (30)Since gcd( m, n ) ∈ N , one obtains with (3) and (4) thatlcm( m ∗ , n ∗ ) = Y i ∈ N + ( p i ) max { q i ( m ∗ ) ,q i ( n ∗ ) } (31) = Y i ∈ N + ( p i ) q i (ˆ n ) − min { q i ( m ) ,q i ( n ) } = h Y i ∈ N + ( p i ) min { q i ( m ) ,q i ( n ) } ! = h (gcd( m, n )) ∈ N ∗ , proving that N ∗ is LCM-closed. Let ( p j ) q ∗ j be a prime power that occurs (as in: divides)in at least one member of N , but not in more than half of them. This means that for atleast half of the members n ∈ N , it holds that q j ( n ) < q ∗ j . But this implies that for atleast half of the members h ( n ) ∈ N ∗ , it holds that q j ( h ( n )) > q (ˆ n ) − q ∗ j ≥
0, meaning that N ∗ has with ( p j ) q (ˆ n ) − q ∗ j +1 an abundant prime power and, therefore, an abundant divisor.2. Up to the point, where N ∗ emerges as GCD-closed, the proof of the second statementfollows in close analogy the one of the first statement. Let now ( p j ) q ∗ j be a prime power inat least half the members of N , but not in all of them. At this point, the question arises,if such a prime power exists, as it – at first sight – could be that all abundant primepowers occurred in all members. However, that this cannot be the case follows from theeasy to check fact that with N , also N / gcd( N ) is LCM-closed, and n n/ gcd( N ) isan LCM-consistent bijection between the two sets. Since Conjecture 1 then also appliesto N / gcd( N ), it is clear that there would be the required type of prime power in this9et, which, by multiplication with the prime power of the same basis (prime) in gcd( N ),would deliver the ( p j ) q ∗ j with the required properties. This means that for at least halfof the members n ∈ N , it holds that q j ( n ) ≥ q ∗ j , implying that for at least half of themembers h ( n ) ∈ N ∗ , it holds that q j ( h ( n )) ≤ q (ˆ n ) − q ∗ j . However, since ( p j ) q ∗ j is nota prime power in all members of N , the prime power ( p j ) q (ˆ n ) − q ∗ j +1 does exist in (as in:divides) members of N ∗ , but in at most half of them. EXAMPLE 12.
This is an example for the first statement in Prop. 1. The GCD-closedset N = { , , , , , , } of Example 1 and 10 has seven elements, and the primepowers 2 , 2 , and 3 are non-abundant. Moreover, lcm( N ) = 24 = 2 ·
3. In Example11, N ∗ = h ( N ) = { , , , , , , } was determined. It can easily be checked that N ∗ is LCM-closed, and that it has the abundant prime powers (divisors) 2 = 2 − ,2 = 2 − , and 3 = 3 − . DEFINITION 3.
Let
N ∈ N + be a non-empty finite GCD-closed (LCM-closed) set.Then N ∗ as in (25) is called its LCM-closed (GCD-closed) dual. Note that (3) and (4) provide the minimum and the maximum of the occurring powerexponents of the prime p i in the members of N as q i (gcd( N )) and q i (lcm( N )). It followswith (24) that the minimum and the maximum of the occurring power exponents of theprime p i in the members of N ∗ are 0 and q i (lcm( N )) − q i (gcd( N )). As was pointed out inthe proof of Proposition 1 for GCD-closed N , observe that for LCM-closed (GCD-closed) N , the set N / gcd( N ) is again LCM-closed (GCD-closed), and( N / gcd( N )) ∗ = N ∗ , (32)since, on the right hand side of (24), the division of N by gcd( N ) simply reduces q i (ˆ n )and q i ( n ) both by q i (gcd( N )), thus – in total – resulting in no change to N ∗ . Therefore,( N ∗ ) ∗ = N (33)holds if and only if gcd( N ) = 1, and, thus, lcm( N ) = lcm( N ∗ ). EXAMPLE 13.
In Example 12, gcd( N ) = gcd( N ∗ ) = 1 and N = ( N ∗ ) ∗ . However, inExample 4, N = { , , , , , , } , (34)and N is LCM-closed, but not GCD-closed, gcd( N ) = 2, and lcm( N ) = 210 = 2 · · · N = { , , , , , , } , (35) N / gcd( N ) = { , , , , , , } , (36) N ∗ = ( N / gcd( N )) ∗ = { , , , , , , } , (37)and, since lcm( N ∗ ) = 105,( N ∗ ) ∗ = N / gcd( N ) = { , , , , , , } . (38) 10 Examples of scope
The sum of the prime power exponents of a natural number n ∈ N + is given by σ PPE ( n ) = X i ∈ N + q i ( n ) . (39)With the findings so far, and returning to the list at the end of Section 3, Conjecture 1for instance holds for an LCM-closed finite set N ⊂ N + that contains at least one elementlarger than 1, if:1. There is an n ∈ N with n = 1, such that the prime factorization of n has a maximumof two prime factors.2. σ PPE (lcm( N ) / gcd( N )) ≤ N ≤
N ≥ σ PPE (lcm( N ) / gcd( N )) .5. N ≤ σ PPE (lcm( N ) / gcd( N ))), if N is such that for any two distinct primepowers (for instance, 2 and 2 are two distinct prime powers in 24) p ′ and p ′′ inlcm( N ) / gcd( N ), there exist m, n ∈ N / gcd( N ) such that p ′ | m , p ′′ | n , but neither p ′′ | m , nor p ′ | n . The long-standing union-closed sets conjecture is known for its applications in graph andlattice theory. This note presents a link of the conjecture to another mathematical field:multiplicative number theory. It would be nice if the here presented equivalent numbertheoretical conjecture or methods turned out to be fruitful in any way.
References [1] Bruhn, H., and O. Schaudt (2015): The Journey of the Union-Closed Sets Con-jecture.
Graphs and Combinatorics (6), 2043–2074.[2] Gauss, C. F. (1801): Disquisitiones Arithmeticae. Gerhard Fleischer, Leipzig.[3] Knill, E. (1991): Generalized degrees and densities for families of sets. PhD thesis,University of Colorado. (Available at: arXiv:math/9411220v1 [math.CO].)[4] Poonen, B. (1992): Union-closed families. Journal of Combinatorial Theory, SeriesA59