Strong Divisibility and Lcm-Sequences

Strong divisibility and lcm-sequences

Andrzej Nowicki

Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, 87-100 Toru´n,Poland, (e-mail: [email protected]).

Abstract Let R be an integral domain in which every two nonzero elements have a greatest

common divisor. Let (an)n>1 be a sequence of nonzero elements in R. We prove that gcd(an, am) = agcd(n,m) for all n, m > 1 if and only if Y an = cd for n > 1, d|n

where c1 = a1 and cn = lcm(a1, a2, . . . , an)/lcm(a1, a2, . . . , an−1) for n > 2. We also present some consequences of this theorem.

1 Introduction

Our goal is to study certain sequences of nonzero elements in an integral domain. Cyclotomic polynomials will play an important role in this article. They posses many interesting divisibility properties. However, similar properties arise in a much broader context that ultimately leads to our main theorem. Consequently, we review cyclotomic polynomials as a motivating example. The nth cyclotomic polynomial is commonly deﬁned by the formula Y Φn(x) = (x − ω), ω where ω ranges over the primitive nth roots of unity. It is well known (see [5] or [13]) that each Φn(x) is a monic irreducible polynomial in Z[x] that satisﬁes the following recursive formula

n Y (1) x − 1 = Φd(x), d|n where d ranges over positive divisors of n.

We now make a brief diversion. Assume that (an)n 1 is a sequence in an abelian group P > G and (bn)n>1 is the sequence defined by bn = d|n ad. Then the Möbiusinversion formula (see [7], [5] or [13]) states that X an = µ(n/d)bd for all n > 1, d|n where µ : N>1 → {−1, 0, 1} is the Möbiusfunction defined as   1 if n = 1, µ(n) = 0 if n is not square-free, k  (−1) if n = p1 ··· pk, with pi distinct primes. If the group law in G is written multiplicatively, then the Möbiusinversion formula takes the following form:

Y Y µ(n/d) (2) bn = ad ⇐⇒ an = bd . d|n d|n We now return to the example of cyclotomic polynomials. Let G be the multiplicative group of the ﬁeld of fractions of Z[x]. Applying the M¨obiusinversion formula to (1), we obtain the equality Y d µ(n/d) Φn(x) = (x − 1) . d|n

This formula may be used as an alternate deﬁnition of Φn(x). Recently, Tomasz Ordowski presented (in a letter to the author) a new alternative deﬁnition of Φn(x). He observed that Φ1(x) = x − 1 and lcm x1 − 1, x2 − 1, . . . , xn − 1 (3) Φn(x) = for n > 2. lcm x1 − 1, x2 − 1, . . . , xn−1 − 1

Tomasz Ordowski suggested also that there exists a similar formula for integer values of Φn(x). In particular, if b > 2 is an integer, then Φ1(b) = b − 1, and lcm b1 − 1, b2 − 1, . . . , bn − 1 (4) Φn(b) = for n > 2. lcm b1 − 1, b2 − 1, . . . , bn−1 − 1

The author presented some proofs of (3) and (4) in [14]. In this article, we show that (3) and (4) are corollaries of our main theorem (Theorem 3.1).

Let R be a unique factorization domain. A sequence (an)n>1 of nonzero elements of R is called a strong divisibility sequence if agcd(m,n) is a greatest common divisor of am and a for all m, n 1. It is well known (see [17], [4] or [3]) that (xn − 1) and (bn − 1) n > n>1 n>1 are strong divisibility sequences in Z[x] and Z, respectively. A sequence (an)n>1 of nonzero elements of R is called special if there exists a sequence (bn)n>1 of nonzero elements in R such that Y an = bd for any n > 1. d|n

Note that if a sequence (an)n>1 is special, then its associated sequence (bn)n>1 is unique. The uniqueness follows from the M¨obiusinversion formula (2) applied to the ﬁeld of Q µ(n/d) fractions of R, namely that bn = d|n ad for any n > 1.

2 It is known (see [19], [11] or [1] p.218, Mathematical Olympiad, Iran 2001) that every strong divisibility sequence in Z is special. Recently, Nathan Bliss, Ben Fulan, Stephen Lovett and Jeﬀ Sommars proved in [3] that the same is true for strong divisibility sequences in arbitrary unique factorization domains. Thus, if (an)n>1 is a strong divisibility sequence in a unique factorization domain R, then there exists a unique sequence (bn)n 1 of nonzero Q > elements in R with b1 = a1 and an = d|n bd for n > 2. The authors of [3] proved also that there exist special sequences which are not strong divisibility sequences.

The main theorem of this article states that if (an)n 1 is a sequence of nonzero elements > Q in R then (an)n>1 is a strong divisibility sequence if and only if an = d|n cd for n > 1, where c1 = a1 and cn = lcm(a1, a2, . . . , an)/lcm(a1, a2, . . . , an−1) for n > 2. The Fibonacci sequence, sequences of the form (an − bn) with a, b ∈ Z, gcd(a, b) = 1, and other well-studied sequences have the strong divisibility property. Applying the main theorem and the M¨obiusinversion formula, we obtain new formulas for these sequences. In particular, we will see in Theorem 3.1 that this type of formula occurs in much more general settings. First, we provide relevant deﬁnitions and preliminary results in Section 2. We prove our main result in Section 3. In Section 4 we provide several examples which illustrate this main result, including immediate corollaries which give the formulas (3) and (4).

2 Gcd-domains and lcm-sequences

Let S be an integral domain, that is, S is a commutative ring with identity without zero divisors. If a, b ∈ S and a 6= 0, then we write a | b if b = ac for some c ∈ S. Let a1, . . . , an be nonzero elements of S.A greatest common divisor (abbreviated gcd) 0 of a1, . . . , an is a nonzero element d in S such that d | ai for i = 1, . . . , n, and if 0 6= d ∈ S 0 0 and d | ai for i = 1, . . . , n, then d | d.A least common multiple (abbreviated lcm) of 0 a1, . . . , an is a nonzero element m in S such that ai | m for i = 1, . . . , n, and if 0 6= m ∈ S 0 0 and ai | m for i = 1, . . . , n, then m | m . Throughout this article we assume that R is a gcd-domain, that is, R is an integral domain and any two nonzero elements in R have a greatest common divisor (see [8]). It follows from the deﬁnition of gcd-domains that if n > 1 and a1, . . . , an are nonzero elements in R, then there exists a greatest common divisor of a1, . . . , an, and moreover, there exists a least common multiple of a1, . . . , an. We adopt the notation (a1, . . . , an) and [a1, . . . , an] for a greatest common divisor and a least common multiple, respectively, of a1, . . . , an. Note that the elements (a1, . . . , an) and [a1, . . . , an] are determined only up to units. This means that if d is a greatest common divisor of a1, . . . , an and w is a least common multiple of a1, . . . , an, then d = u · (a1, . . . , an) and w = v · [a1, . . . , an] for some invertible elements u, v ∈ R. In the remainder of this article, all equalities involving greatest common divisors and least common multiplies hold up to multiplication by a unit. Unique factorization domains, Bezout domains, and valuation domains belong to the class of gcd-domains. Every gcd-domain is integrally closed, and if R is a gcd-domain, then the polynomial ring R[x] is also a gcd-domain (see [8]).

3 We say that two nonzero elements a, b ∈ R are relatively prime if (a, b) = 1. Note that (a, b) is not necessarily a linear combination of a and b. Greatest common divisors and least common multiplies satisfy many divisibility iden- tities. We list only four that are needed for the proof of our main theorem. Proposition 2.1. If a, b, c are nonzero elements of a gcd-domain R, then: (1) [[a, b], c] = [a, [b, c]] = [a, b, c]; (2) [ac, bc] = [a, b]c; (3) if (a, b) = 1 and (a, c) = 1, then (a, bc) = 1; (4) if (a, b) = 1 and a | bc, then a | c. Recall that the equalities which appear in this proposition are determined up to units. If R is a unique factorization domain, then the proof of the above proposition is elementary. A proof for gcd-domains can be found, for example in [8]. The notion of a strong divisibility sequence remains the same in a gcd-domain as in a unique factorization domain, that is, if (an)n>1 is a sequence of nonzero elements of a gcd-domain R, then (an)n>1 is called a strong divisibility sequence if

(am, an) = a(m,n) for all m, n > 1. The following two theorems from [3] will play an important role. They are stated in [3] for unique factorization domains, but they are also valid (with the same proofs) for arbitrary gcd-domains.

Theorem 2.2 ([3]). If (an)n 1 is a strong divisibility sequence of a gcd-domain R, then > Q there exists a unique sequence (bn)n>1 of nonzero elements in R such that an = d|n bd for all n > 1.

Theorem 2.3 ([3]). Let R be a gcd-domain and let (bn)n 1 be a sequence of nonzero Q> elements of R. Let (an)n>1 be the sequence such that an = d|n bd. The sequence (an)n>1 is a strong divisibility sequence if and only if, for all positive integers m and n such that m - n and n - m, the elements bm and bn are relatively prime.

Assume that (an)n>1 is a sequence of nonzero elements of a gcd-domain R. Put

e1 = 1 and en+1 = [en, an] for n > 1.

Then en+1 = [a1, . . . , an] and en | en+1 for all n > 1. Denote by cn the element en+1/en. Then (cn)n>1 is a sequence of nonzero elements in R. We call it the lcm-sequence of (an)n>1. Note that c1 = a1 and

[a1, . . . , an] cn = for all n > 2. [a1, . . . , an−1] Consider the following elementary examples of lcm-sequences in a gcd-domain. If

(an)n>1 is a constant sequence, an = a for all n > 1 with 0 6= a ∈ R, then c1 = a and n cn = 1 for all n > 2. The lcm-sequence of the geometric sequence an = q with 0 6= q ∈ R is equal to the constant sequence cn = q. If R = Z and an = n! for n > 1, then cn = n. The ﬁrst few terms of the lcm-sequence of the triangular numbers tn = n(n + 1)/2 (for n > 1) are: 1, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19.

4 3 The main theorem

The following theorem is the main result of this article.

Theorem 3.1. Let (an)n>1 be a sequence of nonzero elements in a gcd-domain R, and let (cn)n>1 be the lcm-sequence of (an)n>1. Then the following two conditions are equivalent.

(1) (an)n>1 is a strong divisibility sequence. Q (2) an = d|n cd for all n > 1.

Proof. (1) ⇒ (2). Assume that (an)n>1 is a strong divisibility sequence. Then, by Theorem 2.2, there exists a unique sequence (bn)n 1 of nonzero elements in R, such Q > that an = d|n bd for all n > 1. Consider the sequence (en)n>1 deﬁned by e1 = 1 and en+1 = [en, an] for n > 1. We will prove, using an induction with respect to n, that for every n > 1 we have the equality

(5) en+1 = b1b2 ··· bn.

For n = 1 this is obvious because e2 = [e1, a1] = a1 = b1. Now let n > 2 and assume that en = b1 ··· bn−1. Then we have

n−1  Y Y en+1 = [en, an] =  bk, bd = [A · B,A · bn], k=1 d|n where A is the product of the elements bd with d < n and d | n, and B is the product of all elements of the form bd with d < n and d - n. Since (an)n>1 is a strong divisibility sequence, we know by Theorem 2.3 and Proposition 2.1 that for all n the elements B and bn are relatively prime. Hence [B, bn] = Bbn and we have n−1 ! Y en+1 = [AB, Abn] = A[B, bn] = ABbn = bk · bn = b1 ··· bn. k=1 Thus by induction the equality (5) holds for all n > 1. But by the deﬁnition of the lcm- sequence of (an)n>1, c1 = a1 and for n > 2 we have cn = en+1/en = (b1 ··· bn)/(b1 ··· bn−1) = Q Q bn. Hence bn = cn for all n > 1. Therefore an = d|n bd = d|n cd for n > 1. Q (2) ⇒ (1). Assume now that an = d|n cd for n > 1, and let m > 1 be a ﬁxed integer. Denote by U the product of the elements cd with d < m and d | m, and denote by V the product of all elements of the form cd with d < m and d - m. Then we have:

m m−1  Y Y Y UV cm = ci = em+1 = [em, am] =  ci, cd = [UV, Ucm] = U[V, cm]. i=1 i=1 d|m

Hence [V, cm] = V cm and the elements V and cm are relatively prime. Recall that V is the product of all elements cd with d < m and d - m. This implies, by Proposition 2.1, that if d < m and d - m, then the elements cd and cm are relatively prime. We proved that if n, m > 1, n | m and m - n, then the elements cn and cm are relatively prime. Therefore, by Theorem 2.3, the sequence (an)n>1 is a strong divisibility sequence.

5 4 Applications

In this section we provide several examples which illustrate Theorem 3.1, including immediate corollaries which give the formulas (3) and (4) of Section 1. n It is well known (see [17] or [3]) that the sequence an = x − 1 is a strong divisibility sequence in the polynomial ring Z[x]. In this case, we have the equalities n Q x − 1 = d|n Φd(x), where each Φn(x) is the nth cyclotomic polynomial. We know n Q also, by Theorem 3.1, that x − 1 = d|n cd, where (cn)n>1 is the lcm-sequence of (an)n>1. Applying the M¨obiusinversion formula we obtain Φn(x) = cn for all n > 1 and hence we recover Ordowski’s result.

n Corollary 4.1 (Ordowski). Let an = x − 1. Then Φ1(x) = a1 and h i a1, a2, . . . , an−1, an Φn(x) = h i for n > 2. a1, a2, . . . , an−1

In the same manner, we obtain similar formulas for integer values of cyclotomic poly- n nomials. Let b > 2 be an integer and let an = b − 1 for n > 1. Then (an)n>1 is a strong n Q divisibility sequence in Z and b − 1 = d|n Φd(b) for n > 1. Applying Theorem 3.1 and the M¨obiusinversion formula we again obtain that (Φ (b)) is the lcm-sequence of n n>1 (an)n>1. Therefore we have the following corollary.

n Corollary 4.2 (Ordowski). Let b > 2 be an integer and let an = b − 1 for n > 1. Then h i a1, a2, . . . , an−1, an Φn(b) = h i for n > 2. a1, a2, . . . , an−1

Examples and properties of strong divisibility sequences can be found in many books and articles (see for example [9], [16], [3], [10], and [15] pages 150-152). For each such sequence, Theorem 3.1 gives an associated formula connected with its lcm-sequence. We now consider several examples.

Example 4.3. It follows from Corollary 4.2 that

[M1,...,Mn] / [M1,...,Mn−1] = Φn(2),

n where Mn denotes the Mersenne number 2 − 1.

Example 4.4. It is easy to prove and it is well known (see [17] or [4]) that the sequence n n an = x − y is a strong divisibility sequence in the polynomial ring Z[x, y]. In this case, n n Q we have the equalities x −y = d|n Ψn(x, y), where Ψn(x, y) is the polynomial in Z[x, y] deﬁned by ϕ(n) Y d dµ(n/d) Ψn(x, y) = y Φn(x/y) = x − y d|n

6 n n (see for example [12]). Hence, if an = x − y , then

[a1, . . . , an]/[a1, . . . , an−1] = Ψn(x, y) for all n > 2.

n n Example 4.5. If u > v > 1 are relatively prime integers, then the sequence an = u −v is a strong divisibility sequence in Z, and again we have the equalities

[a1, . . . , an]/[a1, . . . , an−1] = Ψn(u, v) for all n > 2.

Example 4.6. The lcm-sequence of the sequence an = n has the form (cn)n>1, where ( p, if n = ps for some prime p and integer s 1, c = > n 1, otherwise.

In this case we have c1 = 1 and cn = Φn(1) for n > 2. Hence [1, 2, . . . , n] = Φ (1) for all n 2. [1, 2, . . . , n − 1] n > Example 4.7. The sequence of repunits, 10n − 1 un = 11 ... 1 = , | {z } 9 n is a strong divisibility sequence in Z. The ﬁrst few terms of the lcm-sequence (cn)n>1 are

1, u2, u3, 101, u5, 91, u7, 10001, 1001001, 9091, u11, 9901, u13, 909091, 90090991.

It is an easy exercise to show that c1 = 1 and cn = Φn(10) for n > 2.

In general, if (an)n>1 is a strong divisibility sequence in a gcd-domain R with the lcm- sequence (c ) , then a | a for all n, and (a /a ) is a strong divisibility sequence in n n>1 1 n n 1 n>1 R with the lcm-sequence (dn)n>1 deﬁned by d1 = 1 and dn = cn for n > 2.

Example 4.8. The sequence of the Fibonacci numbers, F1 = F2 = 1,Fn+2 = Fn+1 + Fn for n > 1, is a strong divisibility sequence in Z (see [18]). The ﬁrst few terms of its lcm-sequence (cn)n>1 are 1, 1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199.

It is not diﬃcult to check that c1 = 1 and√ cn = Ψn(α,√ β) for n > 2, where Ψn(x, y) is 1+ 5 1− 5 deﬁned in Example 4.4 and where α = 2 and β = 2 . Hence

√ √ [F1,F2,...,Fn] 1+ 5 1− 5 = Ψn 2 , 2 [F1,...,Fn−1] for all n > 2.

7 We encourage the reader to investigate the lcm-sequences for the strong divisibility sequences given in the following example, as well as other examples introduced in [3]. Example 4.9. (1). Let (F (x)) be the sequence of Fibonacci polynomials, that is, n n>1 F1(x) = 1,F2(x) = x and Fn+2(x) = xFn+1(x) + Fn(x) for n > 1. It is a strong divisibility sequence in Z[x] (see [6]). (2). The Chebyshev polynomials of the second kind (see [2]) , which satisfy

U0(x) = 1,U1(x) = 2x, Un+2(x) = 2xUn+1 − Un(x), also form a strong divisibility sequence in Z[x] (see [6]). (3). The same holds for the 3-variable polynomials Sn(x, y, z) (see [6]) which satisfy S = 1, S = x and 1 2 ( xSn−1 + ySn−2 for n even, Sn = zSn−1 + ySn−2 for n odd.

Recall (see Section 1) that a sequence (an)n>1 of nonzero elements of R is called special if there exists a (unique) sequence (bn)n 1 of nonzero elements in R such that Q > an = d|n bd for any n > 1. Recall also that every strong divisibility sequence is special. The next example is from [3]. We consider a special sequence which is not a strong divisibility sequence.

Example 4.10. Let an = ϕ(n), where ϕ is the Euler totient function. Since (a4, a6) =

(ϕ(4), ϕ(6)) = (2, 2) = 2 and a(4,6) = a2 = ϕ(2) = 1, the sequence (an)n>1 is not a strong divisibility sequence. But (an)n>1 is a special sequence in Z. The associated sequence (bn)n>1 is deﬁned by   p − 1 if n is a prime number p,  k bn = p if n = p for a prime p and k > 2,  1 otherwise.

Let (cn)n>1 be the lcm-sequence of (an)n>1. The ﬁrst few terms of (cn)n>1 are: 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 11, 1.

Since (an)n>1 is not a strong divisibility sequence, the lcm-sequence of (an)n>1 does not have property (2) in Theorem 3.1. In fact, we have for example 1 = c4 6= b4 = 2 or 2 = c15 6= b15 = 1. We conclude with the following corollary, which is an immediately consequence of Theorem 3.1 and Theorem 3 in [3].

Corollary 4.11. Let (an)n>1 be a strong divisibility sequence in a gcd-domain R. If n > 2 α1 αs has prime factorization n = p1 ··· ps , then

[a1, . . . , an] an = . [a1, . . . , an−1] an/p1 , an/p1 , . . . , an/ps Acknowledgment. The author thanks Mr. Tomasz Ordowski for his interesting formulas. The author thanks also the reviewers for their careful reports with so many clever remarks and very nice ideas.

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