Daisy chains with three generators

Transcription

1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 5 (009), Pages Daisy chains with three generators D. A. Preece Queen Mary University of London School of Mathematical Sciences Mile End Road, London E1 NS U.K. D.A.Preece@qmul.ac.uk and Institute of Mathematics, Statistics and Actuarial Science Cornwallis Building, University of Kent Canterbury, Kent CT 7NF U.K. Abstract For many positive odd integers n, whether prime, prime power or composite, the set U n of units of Z n contains members u, v and w, saywith respective orders ψ, ω and π, such that we can write U n as the direct product U n = u v w. Each element of U n canthenbewrittenin the form u i v j w k where 0 i ψ 1, 0 j ω 1and0 k π 1. We can then often use the structure of u v w to arrange the ψωπ elements of U n in a daisy chain, i.e. in a circular arrangement such that, as we proceed round the chain in either direction, the set of differences between each member and the preceding one is itself the set U n.wedescribe such daisy chains as daisy chains with three generators. Eachsuch daisy chain consists of a succession of super-segments of length ωπ, each made of segments of length π. Within each segment, each successive element is obtained from the preceding one by multiplication by w; within each super-segment, each successive segment is obtained from the preceding one by multiplication by v; each successive super-segment is obtained from the preceding one by multiplication by u. Westudytheexistenceof such arrangements, some of which can be obtained from general constructions which we describe. In many of our examples of the arrangements, one of the generators has order ; if n is prime, that generator must then be 1 (modn), but if n is composite, another square root of 1 (mod n) may occasionally be used.

2 158 D. A. PREECE 1 Introduction Any positive integer n has a prime-power decomposition n = p i q j r k (i, j, k 1) where p,q,r,...are finitely many distinct primes. In standard terminology, the units of the corresponding group Z n are those elements of Z n \0} that are coprime with n. The number of units in Z n is given by Euler s totient function φ(n) =(p 1)p i1 (q 1)q j1 (r 1)r k1 (e.g. [, p. 87]). For n odd, a daisy chain [5] for the units of Z n (in short, a Z n daisy chain) is an ordered arrangement [a 1,a,...,a φ(n) ] of the units on a circle, such that the set of differences b i = a i+1 a i (i =1,,...,φ(n), with a φ(n)+1 = a 1 ) is itself the set of units. Here, as in [5] and elsewhere, we use square brackets to indicate a cycle. However, also as in [5], we replace [ and ] by and when we present a specific daisy chain in a display, and we then replace the commas by spaces. We henceforth use difference to mean a right-minus-left difference of the above form a i+1 a i. Various systematic methods of construction of daisy chains for Z n were given in [5], where the emphasis was on values of n lying in the range 1 <n<00. We now investigate a new type of construction, which proves to be fruitful for both prime and composite values of n. To give a reasonable illustration of the scope of this new type, we now need to extend upwards the range of values of n. For many positive odd integers n, whether prime, prime power or composite, the set U n of units of Z n contains members u, v and w, say with respective orders ψ, ω and π, such that we can write U n = u v w. (Here, as subsequently in this paper, the symbol indicates a direct product.) All, some or none of the orders ψ, ω and π may be prime. Each element of U n can then be written in the form u i v j w k where 0 i ψ 1, 0 j ω 1and0 k π1. We can then often use the structure of u v w to arrange the ψωπ elements of U n in a daisy chain for Z n. We describe such daisy chains as daisy chains with three generators. Eachsuch daisy chain consists of a succession of super-segments of length ωπ, each made up of segments of length π. Within each segment, each successive element is obtained from the preceding one by multiplication by w; within each super-segment, each successive segment is obtained from the preceding one by multiplication by v; each successive super-segment is obtained from the preceding one by multiplication by u. Thus, a daisy chain with three generators is of the following form where, as in [5], a single fence denotes a boundary between two segments within a supersegment, and a double fence denotes a boundary between supersegments:

3 DAISY CHAINS WITH THREE GENERATORS 159 u 0 v 0 w 0 u 0 v 0 w 1... u 0 v 0 w π1 u 0 v 1 w 0 u 0 v 1 w u 0 v 1 w π1 u 0 v ω1 w 0 u 0 v ω1 w 1... u 0 v ω1 w π1 u 1 v 0 w 0 u 1 v 0 w 1... u 1 v 0 w π1 u 1 v 1 w 0 u 1 v 1 w u 1 v 1 w π1 u 1 v ω1 w 0 u 1 v ω1 w 1... u 1 v ω1 w π1. u ψ1 v 0 w 0 u ψ1 v 0 w 1... u ψ1 v 0 w π1 u ψ1 v 1 w 0 u ψ1 v 1 w u ψ1 v 1 w π1 u ψ1 v ω1 w 0 u ψ1 v ω1 w 1... u ψ1 v ω1 w π1 As a daisy chain may be read anticlockwise as well as clockwise, the existence of a particular daisy chain with the three generators u, v and w implies the existence of a daisy chain with the three generators u 1, v 1 and w 1. For this reason, the distinct parameter sets (u, v, w) for any particular n arise in pairs; we use N n to denote the number of such pairs. For prime values of n in the range n<00, Table 1 below gives parameter sets (ψ, ω, π) and(u, v, w) forz n daisy chains with three generators. Table similarly covers prime powers n = p j (j>1), whereas Table covers composite values of n in the range n<10. For each n in Table, we use ξ(n), as in [1], to denote the value of φ(n)/λ(n) where λ(n), which is Carmichael s λ-function [, ], gives the order of an element of maximum order in U n.(ifnis odd and composite, then ξ(n) iseven [1].) Longer versions of Tables 1 and can be viewed [] via the Australasian Journal of Combinatorics website. There, Table 1 is for n<500, and the much longer Table is for n<50. As indicated by the footnotes to Tables 1 and, many of the parameter sets (u, v, w) satisfy simple relationships such as u +v 0(modn) or(u +1)vw 1 (mod n). Mostly, we give only one parameter set (u, v, w) for any particular set (ψ, ω, π) for any particular value of n, but we give two or more sets (u, v, w) ifeach of the listed possibilities satisfies one of the listed simple relationships. Some of these relationships arise from general constructions that can be formalised and proved, as in Sections and below; others arise unexplained. The relationship (u +1)vw 1(modn), satisfied by many examples in Tables 1 and, requires the first element of the second supersegment to be 1 less than the final element of the first supersegment. This situation reflects that of Theorems. and.1 of [5], where the first element of the second segment of a daisy chain is 1 less than the final element of the first segment. However, the relationship (u 1)vw 1 (mod n) also arises for various primes, e.g. n = 79, 1, 57 and 91 (Table 1); then the first element of the second supersegment is 1 more than the final element of the first supersegment.

4 10 D. A. PREECE For composite values of n, the relationships u(v +1)w 1 (mod n) and (u +1)vw 1(modn) frequently occur too. The latter is often satisfied simultaneously with u vw (mod n), in which case we have (u 1)(u +) 0(modn) and (vw 1)(vw +1) 0(modn). Note 1.1: In many of our examples of daisy chains with three generators, one of the generators has order. If n is prime, that generator must then be 1 (modn), but if n is composite, another square root of 1 (mod n) may perhaps be used; we obtain examples of this for n =, 91, 1, 171, 189, 1, 17, 7, 59, 75, 79, 01, 09, 15, 5 and 1 (see Table ). Note 1.: Constructions for Z n daisy chains with two generators were given in [5], but at least three generators are needed if analogous constructions are to be provided for odd composite integers n that have three distinct odd prime factors. For n-values of this last type, we now provide three-generator daisy chains for n = 105, 15, 195, 1, 55, 7, 15 and 5 (again see Table ). Note 1.: A special type of daisy chain for Z n has the property that a i a i+φ(n)/ (mod n) for i =1,,...,φ(n)/. For Z n daisy chains with generators, this property is achieved if ψ is even, with u ψ/ 1(modn). We obtain examples of this for n = 1, 1, 7, 09, 1, and (prime, Table 1), for n =19=1 (Table ) and for n = 15 and 59 (composite, Table ). Having ψ even is not a sufficient condition for the property. Some theorems, n prime We now provide eight theorems that embody general constructions for Z n daisy chains with three generators. Theorems.1. are for prime values of n, andthe- orems.1 are for composite values. Theorem.1 Let n be a prime such that n or 7 (mod 7). Lety be an element of order in Z n, and suppose that there exists an element x, oforder(n 1)/ in Z n, such that x y 1 x i y (mod n) for some i from,,...,(n 1)/}. Then, for Z n, there is a daisy chain with (ψ, ω, π) =((n 1)/,, ) and parameters (u, v, w) =(x, y, 1). Proof: The conditions of the Theorem ensure that ψ, ω and π are mutually prime and that the product u v w is direct. As n 1 (mod ), the element is a quadratic residue modulo n, andasy + y +1 0(modn) wehavey 1 (1 ± ) (mod n). The proposed daisy chain is of the form 1 1 y y (y +1) y +1 x...

5 DAISY CHAINS WITH THREE GENERATORS 11 where the first element in each segment is a quadratic residue modulo n, andthe second is a quadratic non-residue. The difference in the first segment is, which is a quadratic residue, so the set of all the within-segment differences is precisely the set of quadratic residues. The difference at the first single fence is y + 1, and that at the second single fence is 1, and these quadratic non-residues are respectively the sixth and second elements of the first super-segment. So we require the difference xy 1 at the first double fence to equal the fourth element in one of the super-segments. The result follows. Note.1: Theorem.1 fails to yield a daisy chain for n =, but succeeds for n = 7, 19, 11, 8, and 1 (see Table 1) and for n = 99, 57, 571, 18,, 91, 787, 859 and 907. Example.1: n = 7. With y = 9 we may, for example, take x = 9 in Theorem.1, as we then have x y 1 x 7 y (mod n). This gives the following Z 7 daisy chain (see Table 1), where the value x y 1 (mod n) ismarkedwithanasterisk: w v u * Theorem. Let n be a prime with n 7(mod8). Suppose that Z n \0} contains an element x such that the quadratic residues modulo n are the elements of x x +1, withx +1 x +1. Then,forZ n, there is a daisy chain with (ψ, ω, π) = (ord n (x +1), ord n (x), ) and (u, v, w) =(x +1, x 1, 1). Proof: The product u v w is direct as 1 isnotasquareinz n. proposed daisy chain is of the form The

6 1 D. A. PREECE 1 1 x 1 x 1 x x... x x x +1 (x +1) x 1 +1 (x 1 +1) x 1 (x 1 +1) x 1 (x 1 +1)... x(x +1) x(x +1) (x +1).... where the first element in each segment is a quadratic residue modulo n, andthe second is a quadratic non-residue. The difference in the first segment is, which is a quadratic non-residue, so all the within-segment differences are quadratic nonresidues. The differences at the 1 st, nd,...,(ω1) th single fences are x 1 +1, x 1 (x 1 +1),x (x 1 +1),..., namely the initial elements of the nd, rd,..., ω th segments of the second supersegment. So we require the difference x + 1 at the first double fence to be the initial element of a supersegment. The results follows. Note.: The smallest prime for which Theorem. yields a daisy chain is n =11, with x ==5 1 where ord n (x) =5andord n (x + 1) = 1 (see again Table 1). Theorem. Let n be a prime with n (mod8). Suppose that Z n \0} contains an element x, satisfying 1 <x<n, such that Z n \0} = x x +1 = y y +1 where y (x +1) (modn), andwhereord n (x +1) and ord n (y +1) are both even. Then there is a Z n daisy chain (i) with (ψ, ω, π) = (ord n (x), ord n (y), ), (u, v, w) =(x, y 1, 1) ; likewise there is a Z n daisy chain (ii) with (ψ, ω, π) = (ord n (y), ord n (x), ), (u, v, w) =(y, x 1, 1). In each case we have (u +1)vw 1(modn). Proof: The proposed daisy chain (i) is of the form 1 1 y 1 y 1 y y... y y x x xy 1 xy 1 xy xy... xy xy x etc..

7 DAISY CHAINS WITH THREE GENERATORS 1 where the first element in each segment is a quadratic residue modulo n, andthe second is not. The difference in the first segment is, which is a quadratic residue, so all the within-segment differences are quadratic residues. The differences at the 1 st, nd,...,(ω1) th single fences are y 1 (1 + y), y (1 + y),..., namely the second elements of the nd, rd,..., ω th segments of the second supersegment. The difference at the first double fence is x+y = 1, which is the second element of the 1 st segment of the first supersegment. The required result for (i) follows. Note.: The smallest prime for which Theorem. yields daisy chains is n =1 (see Table 1), for which we can take x = 7 where ord n (x) = 11 and ord n (x+1) = 0, with y = 5 where ord n (y) = 15 and ord n (y +1) =. With n<1000, Theorem. yields daisy chains for n = 1,, 5, 57, 571, 59, 79, 87, 859 and 971. Some theorems, n composite We now turn to our theorems for odd composite values of n. Theorems.1,. and., like Theorem.1, provide daisy chains with (ω,π) =(, ). Theorem.1 Let n be a composite odd integer such that divides U n. Suppose that U n contains elements x and y, withord n (x) = U n / and ord n (y) =,such that y + y 1(modn) and U n = x y 1. Suppose further that x y, and that x y 1 x i y (mod n) for some i from,,...,ord n (x)}. Then there is a Z n daisy chain with (ψ, ω, π) =( U n /,, ) and parameters (u, v, w) =(x, y, 1). Proof: If y is a unit of order in Z n where n is composite, y does not necessarily satisfy the condition y + y 1(modn), so the appearance of this condition in the statement of the theorem is not a redundancy (see Theorems. and. below). With this condition imposed, the proof is as for Theorem.1, save that the set of quadratic residues is replaced by the set x y. Note.1: In the range n<50, we can use Theorem.1 for n = 91, 9, 19, 18, 01, 7, 91 and 09 (see Table ). Example.1: n =91=7 1. With y =1,wecantakex = 19 in Theorem.1, as we then have x y 1 x y (mod n). This gives the following Z 91 daisy chain (see Table ), where the value x y 1 (mod n) ismarkedwithanasterisk:

8 1 D. A. PREECE TABLE 1 Parameters for daisy chains with generators n prime (n < 00) n N n ψ ω π Notes Theorem u v w 7 a b b a j j Note Note j continued...

9 DAISY CHAINS WITH THREE GENERATORS 15 TABLE 1 (continued) n N n ψ ω π Notes Theorem u v w j j j j Note Note j j i Notes (all the congruences being interpreted modulo n): a u +v 0 i (u 1)vw 1 b u +v 0 j (u +1)vw 1 TABLE Parameters for daisy chains with generators n aprimepowern = p j with j>1(n<00) n N n ψ ω π u v w i 7 1 (i =1,,...,9or10) i 99 1 (i =1,,...,11 or 1) i 8 1 (i =1,,...,17 or 18)

10 1 D. A. PREECE w v u * supersegments Theorem. Let n =9m where m is an odd integer. Let y =1+m or 1+m, and suppose that U n contains an element x such that ord n (x) = U n / and U n = x y 1. Suppose further that x j y (mod n) and x y + x i (mod n) for some values i and j from,,...,ord n (x)}. Then there is a Z n daisy chain with (ψ, ω, π) =( U n /,, ) and parameters (u, v, w) =(x, y, 1). Proof: The relationships ord n (y) =andy +1 y (mod n) are easily checked. The proposed daisy chain is of the form 1 1 y y (y ) y x... where the first element in each segment comes from x y. The difference in the first segment is, which comes from x y, so the set of all the within-segment differences is precisely the set x y. The difference at the second single fence is +, and that at the first single fence is y+1 y, and these are respectively the sixth and fourth elements of one of the supersegments. So we require the difference xy+ at the first double fence to equal the second element of one of the supersegments. The result follows. Note.: In the range n<50, we can use Theorem. for n = 5,, 99, 117, 15, 171, 07, 5, 1, 79 and. Example. n =99= 11. With y = 7, we can take x = 19, 8, or 7 in Theorem.. Taking x = 19 gives the following Z 99 daisy chain (see Table ), where the value x y +(modn) is marked with a dagger:

11 DAISY CHAINS WITH THREE GENERATORS 17 w v u Theorem. Let n =9m where m is an odd integer. Let y =1+m or 1+m, and suppose that ord n () = U n / and U n = y 1. Then there is a Z n daisy chain with (ψ, ω, π) =( U n /,, ) and parameters (u, v, w) =(, y,1). Proof: The proposed daisy chain is of the form 1 1 y y (y ) (y ) y y (y ) (y ) y y (y ) (y ).. Clearly, the set of within-segment differences is precisely the set of first elements within segments. As (y +1)+(y ) y 0 (mod n) we have y +1 (y ), so the difference at the first single fence equals the final element of the second supersegment. The difference at the second single fence is +, which is the second element of the second supersegment. The difference at the first double fence is y, which is the fourth element of the first supersegment. The required result follows. Note.: In the range n<50, we can use Theorem. for n =, 117 and (see Table ). Example.: n == 7. Taking y = in Theorem. gives the following Z daisy chain (see Table ):

12 18 D. A. PREECE w v u Theorem. Let n be an odd composite integer. Suppose that U n contains an element x such that U n = x x +1 1 and x(x +1)+ 0(modn). Then there are Z n daisy chains with (i) (ψ, ω, π) = (ord n (x), ord n (x +1), ) and (u, v, w) =(x, x +1, 1), and (ii) (ψ, ω, π) = (ord n (x),, ord n (x +1)) and (u, v, w) =(x 1, 1, (x +1) 1 ). Proof: The proposed daisy chain (i) is of the form 1 1 (x +1) (x +1) (x +1) (x +1)... (x +1) 1 (x +1) 1 x x x(x +1) x(x +1) x(x +1) x(x +1)... x(x +1) 1 x(x +1) 1 x x x (x +1) x (x +1) x (x +1) x (x +1)... x (x +1) 1 x (x +1) 1. The difference in the first segment is, which is the first entry in the second segment of the second supersegment. Thus the set of within-segment differences is precisely the set of first entries from each segment. The difference at the first single fence is x + x (mod n), which is the second entry in the first segment of the third supersegment. Thus the set of differences at all the single fences is obtained by taking the second entry from every segment except a segment immediately preceding a double fence. Finally, the difference at the first double fence is x +(x +1) 1 =(x +1) 1 [x(x +1)+1] (x +1) 1 (mod n), which is the final element of the first supersegment. The required result follows. The proposed daisy chain (ii) is of the form

13 DAISY CHAINS WITH THREE GENERATORS 19 1 (x +1) 1 (x +1)... (x +1) 1 (x +1) 1 (x +1)... (x +1) x 1 x 1 (x +1) 1 x 1 (x +1)... x 1 (x +1) x 1 x 1 (x +1) 1 x 1 (x +1)... x 1 (x +1). x x (x +1) 1 x (x +1)... x (x +1) x x (x +1) 1 x (x +1)... x (x +1) x x(x +1) 1 x(x +1)... x(x +1) x x(x +1) 1 x(x +1)... x(x +1). The differences within the first segment are the entries, except the first, in the final segment. The differences at the single fences are the first entries of the first segments in the supersegments. The differences at the double fences are the first entries of the second segments in the supersegments. The required result follows. Note.: Theorem. can be used for n = 59 (see Table ). We now come to an important analogue of Theorem.. Theorem Let n be an odd composite integer such that U n contains an element x with U n = x y 1 where y (x +1) (modn). Suppose that x y. Then (as in Theorem.) thereisaz n daisy chain (i) with (ψ, ω, π) = (ord n (x), ord n (y), ), (u, v, w) =(x, y 1, 1) ; likewise there is a Z n daisy chain (ii) with (ψ, ω, π) = (ord n (y), ord n (x), ), (u, v, w) =(y, x 1, 1). In each case we have (u +1)vw 1(modn). Proof: As for Theorem.. Note : Table includes cases from Theorem. Often, but not always, such cases have uv 1 w and therefore u +v 0(modn). Example : n =. With x =and(u, v, w) =(x, y 1, 1) = (, 10, ), Theorem gives the following Z daisy chain: w v x u

14 170 D. A. PREECE More results for composite n As indicated by the Notes to Table, many of its Z n daisy chains have simple relationships between their parameters u, v and w. Further theorems could be provided to account for the appearance of some of these relationships, but we judge these theorems to be of little interest and to be too numerous to justify their inclusion. Of more interest is a type of relationship that can exist between the parameters for a Z n daisy chain for which n is of the form p i q (p prime, i>1) and those for a Z n daisy chain for n = p i1 q. Thus, for example, we can take (u, v, w) =(10, 8, 1) for n =15= 5, and (u, v, w) =(1, 7, 1) for n =5= 5 (see Table ); here the second set of parameters is obtained by reducing the first set modulo 5. Likewise we can take (u, v, w) = (109, 55, 95) for n =189= 7, and (u, v, w, )= (, 55, ) for n == 7 (again see Table ). However, a set of parameters (u, v, w) forp i q, when reduced modulo p i1 q, does not in general give a valid set of parameters (u, v, w) forp i1 q. Note.1: The Notes to Table also draw attention to special situations that can arise when ψ and ω are both multiples of and π =. We can then sometimes have (a) u ψ/ + v +1 0 (mod n) or (b) u ψ/ + v +1 0 (mod n) in conjunction with uv andw 1(modn). At least one of these situations arises for each of n =, 117, 1, 7, 7 and. 5 More than generators Using obvious definitions, we can extend the concept of daisy chains with three generators to that of daisy chains with n generators where n is an arbitrary positive integer. Computer search has shown that such daisy chains with four generators are not rare. They provide a story for another day. Acknowledgments The author is very grateful to E. R. Vaughan (Queen Mary, University of London) for writing the computer program that produced complete listings of -generator daisy chains, and to the referee who improved two of the theorems and made many helpful comments.

15 DAISY CHAINS WITH THREE GENERATORS 171 TABLE Parameters for daisy chains with generators n composite (n <10) marks a composite n that has distinct prime factors marks a value that generates 1 (modn) n ξ(n) N n ψ ω π Notes Theorem u v w., c b, m , 19 1 c 19 1 t. 1 Note.1(a) b, m r j, s n j, r b, j a c, j Note.1(a) Note Notes (all the congruences being interpreted modulo n): a u +v 0 j (u +1)vw 1 m (u +1)vw 1 q vw u b u vw k u(v +1)w 1 n uv(w 1) 1 r v w c u +v 0 o u v s w 1 d v u +1 p uw v t u continued...

16 17 D. A. PREECE 91 7 TABLE (page ) n ξ(n) N n ψ ω π Notes Theorem u v w c, j b, j 1 1 j n, r n n n j, s t j j b c o o Note continued...

17 DAISY CHAINS WITH THREE GENERATORS 17 TABLE (page ) n ξ(n) N n ψ ω π Notes Theorem u v w., c t. 0 1 Note.1(b) 1 1 n j b, m r n, r k j b j j j j j, s b, m c Note.1(a).1 Note.1(b) Note 1.1 Note 1.1 Note.1(a) Note

18 17 D. A. PREECE References [1] P. J. Cameron and D. A. Preece, Notes on Primitive λ-roots, [] R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 1 ( ), 7. [] R. D. Carmichael, Generalizations of Euler s φ-function, with applications to Abelian groups, Quart. J. Math. (191), [] G.A. Jones and J.M. Jones, Elementary Number Theory. Springer, London, [5] D. A. Preece, Daisy chains a fruitful combinatorial concept, Australas. J. Combin. 1 (008), [] D. A. Preece, Supplementary Tables for Daisy Chains with Three Generators, (Received 7 Aug 008; revised Jan 009)