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A Linear Algebra Approach to the Conjecture of Collatz J.F

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A Linear Algebra Approach to the Conjecture of Collatz J.F View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 394 (2005) 277–289 www.elsevier.com/locate/laa A linear algebra approach to the conjecture of Collatz J.F. Alves, M.M. Graça, M.E. Sousa Dias∗, J. Sousa Ramos Dep. Matemática, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa 1049-001, Portugal Received 17 March 2004; accepted 27 July 2004 Submitted by R.A. Brualdi Abstract We show that a “periodic” version of the so-called conjecture of Collatz can be reformu- lated in terms of a determinantal identity for certain finite-dimensional matrices Mk, for all k 2. Some results on this identity are presented. In particular we prove that if this version of the Collatz’s conjecture is false then there exists a number k satisfying k ≡ 8 (mod 18) for k which the orbit of 2 is periodic. © 2004 Elsevier Inc. All rights reserved. AMS classification: 14G10; 11D04; 15A15; 11D79; 11B83 Keywords: Collatz problem; Periodic orbits; Recurrence; Diophantine linear equations; Congruences; Iter- ation function 1. Introduction The Collatz’s conjecture is a well-known open problem. This is also quoted in the literature as the 3n + 1 problem, the Syracuse problem, Kakutani’s problem, Hasse’s algorithm, and Ulam’s problem. In its traditional formulation the conjecture says that ∗ Corresponding author. E-mail addresses: [email protected] (J.F. Alves), [email protected] (M.M. Graça), [email protected] (M.E. Sousa Dias), [email protected] (J. Sousa Ramos). 0024-3795/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2004.07.008 278 J.F. Alves et al. / Linear Algebra and its Applications 394 (2005) 277–289 any orbit for the iteration function f (n) = (3n + 1)/2, for n odd, and f (n) = n/2, for n even, is always attracted to the value 1. A comprehensive discussion on the sub- ject, namely equivalent formulations and related problems arising in several branches of mathematics, can be found in [4, Chapter 1] and [1,2] where relevant literature is discussed. In this paper we only consider a “periodic” version of the Collatz conjecture which can be stated as: The unique periodic orbit of the function f is the orbit of 1. For convenience, hereafter we refer to this conjecture as being the Collatz’s conjecture. We show how to translate the conjecture of Collatz in terms of the determinant of finite-dimensional matrices denoted by Mk. Namely, in Section 2 we show that the 2 conjecture is true if and only if det Mk = 1 − x ,forallk 2. Using elementary Linear Algebra we prove in Section 3 that det(Mk) = det(Mk−1) 2 for all k/≡ 8 (mod 18). A tentative proof of the equality det Mk = 1 − x ,forall k 2 is outlined. Unfortunately this proof is not complete since we were not able to 2k−1 ≡ prove that a certain tree rooted at 3 , with k 8 (mod 18), has not a vertex equal k to 2 . This tree is constructed in order to produce a nontrivial solution for a certain ˜ homogeneous system Mk−1Y = 0, when k ≡ 8 (mod 18). Let us emphasize that if ˜ such a solution does exist, then det(Mk−1) = 0 and the Collatz’s conjecture is true. Nevertheless, we can prove that such nontrivial solution exists for certain values of k and we give an algorithm for its construction (see Section 4). As a consequence of the discussion on the existence of this solution we show that if the Collatz’s conjecture ≡ k is false then there exist integers k 8 (mod 18) for which the orbit of 2 is periodic. 2. Periodic orbits and a Jacobi formula Consider the following Collatz’s iteration function defined on the set of positive integers N (see [4, p. 11]): 3n+1 = 2 if n is odd, f (n) n (1) 2 if n is even. j For n ∈ N, the orbit of n is the sequence defined as On ={f (n) : j 0},where j j−1 f = f ◦ f is the j-fold iterate of f . An orbit On is said to be periodic, with period p 1, if p is the least positive integer verifying f p(n) = n. For instance, for f given by (1) the orbit of 1 is periodic with period 2 (since f(1) = 2andf 2(1) = 1). A “periodic” version of the so-called Collatz’s conjecture can be stated as: “O1 is the unique periodic orbit of f ”. Note that O1 = O2 ={1, 2}. Throughout this paper each time we refer to the conjecture of Collatz we mean this version of the conjec- ture. For each m 2, consider the following m × m matrix Am, whose entries Am(i, j) are defined by J.F. Alves et al. / Linear Algebra and its Applications 394 (2005) 277–289 279 1iff(i)= j and 1 j m, A (i, j) = (2) m 0otherwise. Remarks (i) Note that for a fixed m 2 the matrix Am does not contain all the informa- tion on the images f(i)for all 1 i m. For instance, when m = 5, one has f(5) = 8 and so the fifth row of A5 is a zero-row. Therefore the orbit of, say 3, can not be traced out from A5,since3→ 5 → 8 → 4 → 2 → 1and8> 5. However the orbit of 4 can be completely followed from the information in A5. (ii) Using the matrix multiplication rule it is easy to see that a power r 1ofAm f r (i) i = ,...,m is related to the values ,for 2 as follows: 1iff r (i) = j andf s(i) m, for 0 <s<r, Ar (i, j) = (3) m 0otherwise. Theorem 1. Let Pm(x) = det(Im − xAm), where Im is the identity matrix of order m and Am given in (2). For f as in (1), the orbit of the point 1 is the unique periodic 2 orbit of f if and only if Pm(x) = 1 − x , for all m 2. Before proving this theorem we will prove the following lemma. Lemma 1. Let A = A be as in ( ). Then, (I − xA) = − x2 if and only if m 2 det m 1 0 if p odd, Tr(Ap) = (4) 2 if p even. Proof. Let A be a square real matrix. Recall that as a consequence of Jacobi’s for- mula for the derivative of a determinant (see for instance [3, p. 570]), one has the well known identity between formal power series in the indeterminate x Tr(An) 1 exp xn = . (5) n det(Im − xA) n1 2 If det(Im − xA) = 1 − x , then from (5)weget Tr(An) 1 exp xn = , (6) n 1 − x2 n1 or equivalently Tr(An) xn =−log(1 − x2). (7) n n1 Differentiating (7) we obtain − 2x Tr(An)xn 1 = = 2x + 2x3 + 2x5 +··· (8) 1 − x2 n1 So, comparing both members of Eq. (8)wegettheresultin(4). 280 J.F. Alves et al. / Linear Algebra and its Applications 394 (2005) 277–289 Conversely, from (4)wehave(8). Integrating both members of (8) we obtain (7) and equivalently (6). Finally, comparing (6) with (5), we have det(Im − xA) = 1 − x2. Proof (of Theorem 1). Suppose there exists an orbit of period p that is not the orbit p of 1, say Oa with a/= 2. That is, we are assuming that f (a) = a. Let us choose m = max Oa and Am defined as in (2). p From the expression of the powers of Am,givenin(3), we have A (a, a) = 1. p p Then, as a>2, we have Tr(Am) 3ifp is even, and Tr(Am) 1ifp is odd (note p = p = that Tr(A2 ) 2ifp is even and Tr(A2 ) 0ifp is odd). So, by Lemma 1 we get 2 Pm(x) = det(Im − xAm)/= 1 − x which is a contradiction. Conversely, suppose that the orbit of 1 is the unique periodic orbit. By definition of Am one has n = ∈{ }: n = i = Tr(Am) # k 1,...,m f (k) k and f (k) m, for all i 1,...,n , n = n = thus Tr(Am) 2, if n is even and Tr(Am) 0, if n is odd. Therefore by Lemma 1 2 we get det(Im − xA) = 1 − x for all m 2. 3. On the determinant of a Collatz matrix The aim of this section is to discuss whether Mk = Ik + xAk satisfies the iden- 2 tity det(Mk) = 1 − x ,forallk 2. We will call the matrix Mk a Collatz matrix. 2 By Theorem 1, the equality det(Mk) = 1 − x ,forallk 2, is equivalent to the Collatz’s conjecture. The Collatz matrix Mk = (mij )i,j=1,...,k is given by 1ifi = j, = = mij x if f(i) j, 0otherwise, that is 1 x 000··· m1k x 1000··· m 2k 0010x ··· m 3k 0 x 010··· m Mk = 4k . (9) ··· 00001 m5k . . mk1 mk2 mk3 mk4 mk5 ··· 1 J.F. Alves et al. / Linear Algebra and its Applications 394 (2005) 277–289 281 The following lemma presents some fundamental properties of the Collatz matrix. Lemma 2. The Collatz matrix Mk has the following structure: (a) There is no more than one value x per row. (b) A column j(2 j k) of Mk has one value x above the main diagonal, if and only if j ≡ 2 (mod 3). (10) In this case x belongs to an odd row of Mk.
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