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
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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 JOHN G. STEVENS AND DIANA M. THOMAS DEPARTMENT OF MATHEMATICAL SCIENCES MONTCLAIR STATE UNIVERSITY UPPER MONTCLAIR, NJ Abstract. Ths artcle consders the n-number Ducc game as a lnear map over the n dmensonal vector space Z n 2. Perod lengths are characterzed as orders of factors of the mnmal polynomal of that transformaton. The mnmal polynomal can be explctly obtaned for any n. condtons, ntally found by Ehrlch. We provde algebrac proofs of dvsblty In addton, all perodc cycle lengths for 3» n» 40 are provded. 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) =(jx 1 x 2 j; jx 2 x 3 j;::: ;jx n x 1 j) (1) 1
2 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 6= 2 k, t has been proved that every ntal vector converges to a perodc cycle [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 lengths were calculated only to n =165. Ths artcle wll develop new nsghts nto the perod lengths for any postventeger n by consderng the Ducc game as a map on the vector space Z n 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 observaton s due to an early result that states every ntal vector converges n a fnte number of teratons to a perodc soluton of the form k(x 1 ;x 2 ;::: ;x n ) where x 2 f0; 1g and k s a postve constant [3]. 2
3 Ehrlch notced that the Ducc map on bnary vectors can be wrtten as Dx =((x 1 + x 2 ) mod 2; (x 2 + x 3 ) mod 2;::: ;(x n + x 1 ) mod 2); whch s clearly a lnear map on Z n 2. The matrx representaton of D n the standard bass s gven by, A = ::: ::: ::: ::: C A = I + S L (2) where S L s the left shft map on Z n 2. Usng the formulaton (2), Ehrlch 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 attempted 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 dvsblty condtons by usng the algebrac structure of the Ducc map on Z 2. The followng defntons wll be used extensvely n our analyss. 3
4 Defnton 2.1. The mnmal annhlatng polynomal of a vector v 2 Z n 2 s the monc polynomal μ v ( ) of least degree such that μ v (A)v =0. 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) 6= 0. Then the order of μ v ( ), ord(μ v ( )), s defned to be the smallest natural number, c, suchthatμ v ( )j c 1. If μ v (0) = 0, then μ v ( ) can be wrtten as k ~μ v ( ), for some postve nteger k, where the polynomal, ~μ v ( ), has the property, ~μ v (0) 6= 0. In ths case, the order of μ v ( ) s defned to be the order of ~μ v ( ). A characterzaton of perod lengths by Rchman for odd n based on orders of polynomals was quoted n [16]. To the best of our knowledge, the paper contanng the proof of ths result never appeared. We now provde a general characterzaton for any postve nteger n, and any lnear map. Ths result was ntally proved n [25] to study a smlar lnear map on Z n 2. Theorem 2.1. Let v 2 Z n 2. Let μ v( ) be the mnmal annhlatng polynomal of v. Assume that μ v ( ) = k ~μ v ( ) where k 0 and ~μ v ( ) s a monc polynomal wth ~μ v (0) 6= 0. Then the k th terate of v belongs to a perodc cycle wth perod length c = ord(μ v ). Proof: 4
5 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( ) = j ( c 1) has the property that p(a)v =0. Snce the mnmal polynomal dvdes any other annhlatng polynomal, t follows that μ v ( )j j ( c 1) (3) Usng the assumpton that μ v ( ) = k ~μ v ( ), yelds that k j j and ~μ v ( )j c 1. Wewllnow showthatord(~μ v ( )) must equal c. To see ths, assume on the contrary that ord( ~μ v ( )) = l for some natural number l < c. Ths means that ~μ v ( )j l 1 whch yelds, k ( l 1) = μ v ( )q( ) 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, gvng a contradcton snce the perod of the cycle that v converges to s c. Thus, c = ordμ v ( ). Next we wll prove that k = j. Snce k j j, k» j. We wll show that k cannot be strctly less than j. To see ths assume on the contrary that k<j. Now ~μ v ( )j c 1 5
6 by the defnton of order. Therefore, μ v ( )j k ( c 1) and so k ( c 1) = μ v ( )μq( ); for some polynomal, μq( ). From the defnton of mnmal annhlatng polynomal, A k (A c 1)v = μq(a)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<js false and k cannot be strctly less than j. Ths shows that k must equal j. 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 the mnmal polynomal of A, μ n ( ). We do so by frst computng the characterstc polynomal of A. The structure of the matrx A I provdes some mportant observatons: A I = ::: ::: ::: ::: C A (4)
7 The n 1 st mnor determnant ofa I s equal to one. Therefore, the characterstc polynomal s p n ( ) =(1 ) n +1: A result n [13] states that μ n ( ) = p n ( )[q n 1( )] 1 where q n 1( ) s the greatest common factor of the n 1 rowed mnor determnants of A I. Snce we already know that one of the n 1 mnor determnants s equal to one, μ n ( ) =p n ( ) =(1 ) 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 ( ) = (1 + ) n +1: Moreover, for n odd, μ n ( ) = ~μ v ( ) and so any v that converges to the maxmal cycle does so n at most one teraton. Defne c 1 =2 j 1 where j s the order of 2 modulo n. If nj2 l +1,forsome l, then let m = mnfl : nj2 l +1g 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 cjc 1 and cjc 2 f c 2 exsts. We now provde alternate algebrac proofs of ths dvsblty condton. The structure 7
8 of these arguments gve nsght nto the connecton between Ehrlch's results and the mnmal polynomal of A. Proposton 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 ( ) are smple, a result from [14] states that c =ord(μ n ( )) = mn s z s =1; where z s arootofμ n ( ). Now f z s a root of μ n ( ) then (1 + z ) n =1: Let x =1+z. Then x n =1and 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 cjc 1. To see ths, observe that (1 + x ) c 1 = (1 + x ) 2j 1 = 1+x + :::x 2j 1 = 1+x2j : 1+x 8
9 We know x 2j 1 =1,snce nj2 j 1. Therefore, x 2j 1+x 2j 1+x =1: = x and so Proposton 2.4. Let n be odd and suppose that c 2 exsts. Then cjc 2. Proof: Smlar to Proposton 2.3, we compute (1 + x ) c 2, Snce x 2m +1 =1,x 2m = x 1 (1 + x ) c 2 = (1 + x ) 2m 1 Λ n so Λ = 1+x + :::+ x 2m 1 n» m 1+x 2 n = 1+x» m 1+x 2 n» = x n x +1 n =1: 1+x 1+x Ths proves cjc 2. The next lemma relates c 2 to c 1 when c 2 exsts. Proposton 2.5. Let n be odd and suppose that c 2 exsts. Then c 2 jc 1. 9
10 Proof: We wll frst show that j =2m. Snce 2 m 1( mod n), 2 2m 1( mod n), whch mples that j» 2m. Now, f j < m, then 2 m j 1( mod n), contradctng the mnmaltyofk. Moreover, by defnton, m 6= j, and f m<j< 2m, then2 j m 1( mod n), agan contradctng the mnmalty ofm. Hence j = 2m as clamed. It follows that, c 1 =(2 m + 1)(2 m 1) s dvsble by c 2. Ehrlch provded four examples to show thatc 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, obtaned usng Theorem Perods of the n-number Ducc game. In addton to cycles wth the maxmal perod c, there can exst cycles wth shorter perods that are proper dvsors of c. If g( ) s a proper dvsor of ~μ( ), then there exsts a vector wth g( ) as ts mnmal annhlatng polynomal. As Theorem 2.1 states, the vector s on a cycle wth perod equal to the order of g( ). In fact, all possble perods can be obtaned by examnng ~μ and all of ts dvsors. 10
11 For example, for n =17,~μ( ) =g 1 ( )g 2 ( ) where, g 1 ( ) = and g 2 ( ) = : The order of g 1 s 85 and the order of g 2 s 255. Although the majorty of cycles are of length 255, there do exst three cycles of length 85. The algorthm to obtan the complete cyclc structure (state dagram) for a lnear map on Z n p based on ths approach s gven n [25]. 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, 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 6= c 1 ;c 2? The even case has been looked at but not as extensvely as the odd case. Are there smlar dvsblty condtons connected to the mnmal polynomal n the even case? We beleve that the algebrac structure may provde some answers to these questons. 11
12 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 =22 3 1; 341; 682 n =4 1 1 n =23 2 1; 2047 n =5 2 1; 15 n =24 5 1; 3; 6; 12; 24 n =6 3 1; 3; 6 n =25 3 1; 15; n =7 2 1; 7 n =26 3 1; 819; 1638 n =8 1 1 n =27 4 1; 3; 63; n =9 3 1; 3; 63 n =28 4 1; 7; 14; 28 n =10 3 1; 15; 30 n = n =11 2 1; 341 n =30 7 1; 3; 5; 6; 10; 15; 30 n =12 4 1; 3; 6; 12 n =31 2 1; 31 n =13 2 1; 819 n = n =14 3 1; 7; 14 n =33 4 1; 3; 341; 1023 n =15 4 1; 3; 5; 15 n =34 5 1; 85; 170; 255; 510 n = n =35 6 1; 7; 15; 105; 819; 4095 n =17 3 1; 85; 255 n =36 7 1; 3; 6; 12; 63; 126; 252 n =18 5 1; 3; 6; 63; 126 n =37 2 1; n =19 2 1; 9709 n =38 3 1; 9709; n =20 4 1; 15; 30; 60 n =39 6 1; 3; 455; 819; 1365; 4095 n =21 5 1; 3; 7; 21; 63 n =40 5 1; 15; 30; 60; 120 Table 1. Perod lengths under teratons of the Ducc map. Acknowledgments The authors would lke to thank Marc Chamberland for generatng nterest n the problem and provdng us wth lterature. 12
13 References 1. O. Andrychenko and M. Chamberland Iterated Strngs and Cellular Automata, Mathematcal Intellgencer, 22(4):33 36, (2000). 2. F. Breuer. A Note on a Paper by Glaser and Schöff,Fbonacc Quarterly, 36(5): , (1998). 3. M. Burmester, R. Forcade and E. Jacobs. Crcles of Numbers, Glasgow Math. J. 19: ,(1978). 4. L. Carltz and R. Scovlle, n Solutons", SIAM Revew, 12: , (1970). 5. M. Chamberland, Unbounded Ducc Sequences, Journal of Dfference Equatons, to appear. 6. C. Camberln and A. Marengon. Su una nteressante curost, t a numerca Perodche d Matematche, 17:25 30, (1937). 7. J. Creely. The Length of a Three-Number Game, Fbonacc Quarterly, 26: , (1988). 8. A. Ehrlch. Perods n Ducc's n-number Game of Dfferences, Fbonacc Quarterly, 28: , (1990). 9. Freedman B. Freedman. The Four Number Game, Scrpta Math., 14:35 47, (1948). 10. H. Glaser and G. Schöffl. Ducc-Sequences and Pascal's Trangle, Fbonacc Quarterly, 33: , (1995). 11. J.M. Hammersley. n Problems" SIAM Revew, 11:73 74, (1969). 13
14 12. R. Honsberger, Ingenuty n Mathematcs, Yale Unversty, (1970). 13. Jacobson, N., Lectures n Abstract Algebra, Volume II-Lnear Algebra (1953). 14. Ldl R., Nederreter H., Fnte Felds, Encyclopeda of Mathematcs and ts Applcatons, 20, (1983). 15. M. Lotan, A Problem n Dfference Sets, Amercan Mathematcal Monthly, 56: , (1949). 16. A. Ludngton Furno. Cycles of Dfferences of Integers, Journal of Number Theory, 13: , (1981). 17. A. Ludngton. Length of the 7-Number Game, Fbonacc Quarterly, 26: , (1988). 18. A. Ludngton-Young. Length of the n-number Game, Fbonacc Quarterly, 28: , (1990). 19. A. Ludngton-Young. Ducc-Processes of 5-tuples, Fbonacc Quarterly, 36(5): , (1998). 20. A. Ludngton-Young. Even Ducc-Sequences, Fbonacc Quarterly, 37(2): , (1999). 21. McLean K.R. McLean. Playng Dffy wth real sequences, Mathematcal Gazette, 83:58 68, (1999). 22. Mller R. Mller. A Game wth n Numbers, Amercan Mathematcal Monthly, 85: , (1978). 14
15 23. Pompl F. Pompl. Evoluton of fnte sequences of ntegers ::: Mathematcal Gazette, 80: , (1996). 24. I.R. Sprague, Recreaton n Mathematcs, Dover, (1963). 25. Stevens, John G., On the constructon of state dagrams for cellular automata wth addtve rules., Informaton Scences 115: 43 59,(1999). 26. B. Thwates. Two Conjectures or how to wn $1100, Mathematcal Gazette, 80:35 36, (1996). 27. W. Webb, The Length of the Four-Number Game, Fbonacc Quarterly, 20:33 35, (1982). 28. F.-B. Wong, Ducc Processes, Fbonacc Quarterly, 20:97 105, (1982). 29. P.Zvengrowsk, Iterated Absolute Dfferences, Mathematcs Magazne,52(1):36 37, (1979). 15
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