A CHARACTERIZATION FOR THE LENGTH OF CYCLES OF THE N - NUMBER DUCCI GAME

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1 A CHARACTERIZATION FOR THE LENGTH OF CYCLES OF THE N - NUMBER DUCCI GAME NEIL J. CALKIN DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC AND JOHN G. STEVENS AND DIANA M. THOMAS DEPARTMENT OF MATHEMATICAL SCIENCES MONTCLAIR STATE UNIVERSITY UPPER MONTCLAIR, NJ Abstract. Ducc n-number games have been studed over the last century by a varety of researchers. Ths artcle consders the Ducc Game as a map over the fnte feld Z 2. The perod lengths are characterzed through orders of mnmal annhlatng polynomals. The mnmal polynomal of the map can be explctly wrtten down for any n. Usng the mnmal polynomal, we provde algebrac proofs of dvsblty condtons, ntally obtaned n Ehrlch. In addton, data on all perodc cycles for vector lengths up to n = 40 are computed. 1. Introducton In the late 1800 s, E. Ducc made a seres of observatons on teratons of the map D : Z n Z n, D(x) = ( x 1 x 2, x 2 x 3,..., x n x 1 ) (1) where x = (x 1, x 2,..., x n ) [12]. The dynamcs of the Ducc map have been examned for specal cases of n n [7,17,27]. In addton, many nterestng results have been developed for arbtrary n n [1,2,8]. One of the man results, whch has been proved several tmes n the lterature, states that for n = 2 k for some postve nteger k, all ntal vectors converge to the zero vector [1,5]. For the case n 2 k, t has been proved that every ntal vector converges to a perodc orbt [8]. Specfc propertes on the lengths of the perod have been examned by Ehrlch n [8]. Ehrlch proved some dvsblty condtons relatng odd vector length to maxmal perod length. Usng these relatonshps, he generated maxmal perod lengths for odd n. Due to computng lmtatons, the data was only calculated to n = 165. Ths artcle wll develop new ntuton nto the perod lengths for any postve nteger n by consderng the Ducc game as a map on the vector space Z n 2. 1

2 2. The n number Ducc as a map on the vector space Z n 2 In order to understand the dynamcs of the Ducc map on Z, we need only understand how the map behaves on bnary vectors. Ths s due to an early result statng every ntal vector converges to a perodc soluton of the form k(x 1, x 2,..., x n ) where x {0, 1} and k s a postve constant n fnte tme [3]. Ehrlch observed that the Ducc map on bnary vectors can be wrtten as Ax = ((x 1 + x 2 ) mod 2, (x 2 + x 3 ) mod 2,..., (x n + x 1 ) mod 2), whch s a lnear map on Z2 n. In fact, n matrx form A = = I + S L (2) where S L s the left shft map on Z n 2. Usng the formulaton (2), Ehrlch neatly proves that all vectors converge to the zero vector for n = 2 k. One smply expands (I +S L ) 2k usng the bnomal theorem. Snce all the nner bnomal coeffcents are multples of two, (I + S L ) 2k = I + S n L = I + I = 0. Ehrlch strove to fnd a formula for the maxmal perod length for odd n. He was unable to dscover a concse general formula but he was able to prove some valuable dvsblty relatonshps between vector length and the sze of the maxmal perod. We wll reprove hs dvslbly condtons by usng the algebrac structure of the Ducc map on Z 2 by consderng the Ducc map as a lnear map on the vector space, Z n 2 and usng lnear algebra to obtan a formula for the lengths of all cycles for any postve nteger n. The followng defntons wll be used extensvely n our analyss. Defnton 2.1. The mnmal annhlatng polynomal of a vector v Z2 n monc polynomal µ v (λ) of least degree such that µ v (A)v = 0. s the The exstence of such a polynomal s guaranteed by the Cayley-Hamlton theorem whch states that the characterstc polynomal of A wll annhlate A. Defnton 2.2. Suppose that µ v (0) 0. Then the order of µ v (λ), ord(µ v (λ)), s defned to be the smallest natural number, c, such that µ v (λ) λ c 1. If µ v (0) = 0, then µ v (λ) can be wrtten as λ k µ v (λ), for some postve nteger k, where the polynomal, µ v (λ), has the property, µ v (0) 0. In ths case, the order of µ v (λ) s defned to be the order of µ v (λ). A characterzaton of perod lengths by Rchman for odd n based on orders of polynomals was quoted n [16]. Unfortunately, the paper contanng the proof of ths result never appeared. We now provde a general characterzaton for any postve 2

3 nteger n, and any lnear map. Ths result was ntally proved n [25] to study a smlar lnear map on Z 2. Theorem 2.1. Let v Z n 2. Let µ v (λ) be the mnmal annhlatng polynomal of v. Assume that µ v (λ) = λ k µ v (λ) where k 0 and µ v (λ) s a monc polynomal wth µ v (0) 0. Then the k th terate of v belongs to a perodc cycle wth perod length c = ord(µ v ). Proof: Let A j v be the frst terate that belongs to the perodc cycle. Denote the length of the cycle by c. Then by defnton of perodcty, A c (A j v) = A j v A c (A j v) A j v = 0 A j (A c I)v = 0 Therefore, the polynomal p(λ) = λ j (λ c 1) has the property that p(a)v = 0. Snce the mnmal polynomal dvdes any other annhlatng polynomal, t follows that µ v (λ) λ j (λ c 1) (3) Usng the assumpton that µ v (λ) = λ k µ v (λ), yelds that λ k λ j and µ v (λ) λ c 1. We wll now show that ord( µ v (λ)) must equal c. To see ths, assume on the contrary that ord( µ v (λ)) = l for some natural number l < c. Ths means that µ v (λ) λ l 1 whch yelds, λ k (λ l 1) = µ v (λ)q(λ) for some polynomal q. Therefore, A k (A l I)v = µ v (A)q(A)v = q(a)µ v (A)v = 0. It follows that A l A k v = A k v and so A k v s on a perodc cycle of length l. Ths creates a contradcton because the perod of the cycle that contans v s c. Ths proves that c = ordµ v (λ). Next we wll prove that k = j. Snce λ k λ j, k j. We wll show that k cannot be strctly less than j. To see ths assume that k < j. Now µ v (λ) λ c 1 by defnton of order. Therefore, µ v (λ) λ k (λ c 1) and so λ k (λ c 1) = µ v (λ). By defnton of mnmal annhlatng polynomal, A k (A c 1)v = µ v (A)v = 0. Therefore, A c (A k v) = A k v and so A k v s on the perodc cycle. But A j v s the frst terate on the cycle. Hence our assumpton that k < j s false and k cannot be strctly less than j. Ths shows that k must equal j and completes the proof. Snce there always exsts a vector whose mnmal annhlatng polynomal s the mnmal polynomal, the perod of the maxmal cycle s equal to the order of the mnmal polynomal. Therefore, t wll be useful to obtan the exact formulaton of 3

4 the mnmal polynomal of A, µ n (λ). We do so by frst computng the characterstc polynomal of A. The structure of the matrx A λi provdes some mportant observatons: 1 λ λ A λi =... (4) λ λ The n 1 st mnor determnant of A λi s equal to one. Therefore, the characterstc polynomal s p n (λ) = (1 λ) n + 1. A result n [13] states that µ n (λ) = p n (λ) [q n 1 (λ)] 1 where q n 1 (λ) s the greatest common factor of the n 1 rowed mnor determnants of A λi. Snce we already know that one of the n 1 mnor determnants s one, µ n (λ) = p n (λ) = (1 λ) n + 1. Snce we are workng over a feld of characterstc 2, combnng these observatons wth Theorem 2.1 yelds the followng corollary, Corollary 2.2. Let n be a postve nteger. Then the perod of the maxmal cycle under A s equal to the order of the mnmal polynomal of A, µ n (λ) = (1 + λ) n + 1. Moreover, for n odd, µ n (λ) = λ µ v (λ) and so any v that converges to the maxmal cycle converges n at most one step. Defne c 1 = 2 j 1 where j s the order of 2 modulo n. If n 2 l + 1, for some l, then let m = mn{l : n 2 l + 1} and defne c 2 = n(2 m 1). Note that the exstence of c 1 s always guaranteed by Euler s Theorem. Ehrlch proved that c c 1 and c c 2 f c 2 exsts. We now provde alternate algebrac proofs of ths dvsblty condton that s based on cyclotomc felds. The structure of these proofs gve nsght nto the connecton between Ehrlch s results and the mnmal polynomal of A. Lemma 2.3. For odd n the perod of the maxmal cycle, c dvdes c 1. Proof: Snce the roots of the mnmal polynomal, µ n (λ) are smple, a result from [14] states that c = ord(µ n (λ)) = mn s z s = 1, where z s a root of µ n (λ). Now f z s a root of µ n (λ) then (1 + z ) n = 1. 4

5 Let x = 1 + z. Then x n = 1 and z = 1 + x. Therefore, we seek, mn s (1 + x ) s = 1 where x s an n th root of unty. We wll show that (1 + x ) c 1 = 1 whch wll prove that c c 1. To see ths observe that (1 + x ) c 1 = (1 + x ) 2j 1 = 1 + x +... x 2j 1 But we know x 2j 1 Ths completes the proof. = 1 + x2j. 1 + x = 1 snce n 2 j 1. Therefore x 2j 1 + x 2j 1 + x = 1. = x and so Lemma 2.4. Let n be odd and suppose that c 2 exsts. Then c c 2. Proof: Smlar to Lemma 2.3, we compute (1 + x ) c 2, Snce x 2m +1 Ths proves c c 2. = 1, x 2m (1 + x ) c 2 = [ (1 + x ) 2m 1 ] n = x 1 so [ 1 + x 2 m 1 + x = [ 1 + x x 2m 1 [ ] 1 + x 2 m n = 1 + x ] n = x n [ ] n x + 1 = x The next lemma relates c 2 to c 1 when c 2 exsts. Lemma 2.5. Let n be odd and suppose that c 2 exsts. Then c 2 c 1. Proof: We wll frst show that j = 2m. To see ths observe that 2 2m ( 1) 2 1( mod n), so that j 2m. Now, f j < m, then 2 m j 1 1( mod n), contradctng the mnmalty of k. Moreover, by defnton, m j, and f m < j < 2m, then 2 j m 1( mod n), agan contradctng the mnmalty of m. Hence j = 2m as clamed. 5 ] n

6 It follows that, c 1 = (2 m + 1)(2 m 1) s dvsble by c 2. Ehrlch provded four examples to show that c does not necessarly have to equal c 1 or c 2, namely n = 37, 95, 101 and 111. Obvously, the maxmal perod n these cases was a proper common dvsor of c 1 and c 2 and s also a multple of n. The next secton provdes data for cycles of the Ducc map up to n = 40, usng Theorem Perods of the n-number Ducc game. An algorthm to obtan the orders of the mnmal annhlatng polynomals for a lnear map on Z 2 was developed n [25]. All perods are techncally the orders of the rreducble factors of the mnmal polynomal. Usng Maple, the rreducble factors can be effcently determned through the Smth Normal form of A. The output of ths procedure appled to the Ducc map for vector lengths up to n = 40 are provded n Table 1. In addton to the nformaton n Table 1, the program also generates the number of vectors n each cycle and the maxmum number of teratons needed to arrve n the cycle and the rreducble factors of the mnmal polynomal. We note that fxed ponts are consdered a cycle of length one. Many nterestng questons reman on the Ducc map. For example, s there a way to predct when c = c 1 or c 2? If so, s there a method of determnng the perod for the cases when c c 1, c 2? The even case has been looked at but not as extensvely as the odd case. Are there smlar concse dvsblty condtons connected to the mnmal polynomal n the even case? We beleve that the algebrac structure may lend some answers to these questons. Acknowledgements The authors would lke to thank Marc Chamberland for generatng nterest n the problem and provdng us wth lterature. 6

7 Vector Length Number of Cycles Cycle Lengths Vector Length Number of Cycles Cycle Lengths of Dfferent Lengths of Dfferent Lengths n = 3 2 1, 3 n = , 341, 682 n = n = , 2047 n = 5 2 1, 15 n = , 3, 6, 12, 24 n = 6 3 1, 3, 12 n = , 15, n = 7 2 1, 63 n = , 819, 1638 n = n = , 3, 63, n = 9 3 1, 3, 63 n = , 7, 14, 28 n = , 15, 30 n = 29 n = , 341 n = , 3, 5, 6, 10, 15, 30 n = , 3, 6, 12 n = , 31 n = , 819 n = n = , 7, 14 n = , 3,, 341, 1023 n = , 3, 5, 15 n = , 85, 170, 255, 510 n = n = , 7, 15, 105, 819, 4095 n = , 85, 255 n = , 3, 6, 12, 63, 126, 252 n = , 3, 6, 63, 126 n = 37 n = , 9709 n = , 9709, n = , 15, 30, 60 n = , 3, 455, 819, 1365, 4095 n = , 3, 7, 21, 63 n = , 15, 30, 60, 120 Table 1. Perod lengths under teratons of the Ducc map. 7

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