Frobenius numbers of generalized Fibonacci semigroups
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1 Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC , USA gmtthe@clemson.edu Received:, Accepted:, Published: Abstrct The numericl semigroup generted by reltively prime positive integers 1,..., n is the set S of ll liner combintions of 1,..., n with nonnegtive integrl coefficients. The lrgest integer which is not n element of S is clled the Frobenius number of S. Recently, J. M. Mrín, J. L. Rmírez Alfonsín, nd M. P. Revuelt determined the Frobenius number of Fiboncci semigroup, tht is, numericl semigroup generted by certin set of Fiboncci numbers. In this pper, we consider numericl semigroups generted by certin generlized Fiboncci numbers. Using technique of S. M. Johnson, we find the Frobenius numbers of such semigroups obtining the result of Mrín et. l. s specil cse. In ddition, we determine the duls of such semigroups nd relte them to the ssocited Lipmn semigroups. 1. Introduction Given set of reltively prime positive integers 1,..., n, let S denote the set of liner combintions of 1,..., n with nonnegtive integrl coefficients. Since 1,..., n re reltively prime, every sufficiently lrge integer N is n element of S. The lrgest integer which is not n element of S is clled the Frobenius number of S nd is denoted by g(s). The Frobenius problem is to determine g(s). An excellent generl reference on the Frobenius problem is [11]. In discussing the Frobenius problem, it is convenient to use the terminology of numericl semigroups. The set S defined bove is clled the numericl semigroup generted by 1,..., n nd is denoted by S = 1,..., n ; tht is, { n 1,..., n := c i i : c i N where N denotes the set of nonnegtive integers. Typiclly, we ssume tht i=1 i / 1,..., i 1, i+1,..., n 1 This work ws supported in prt by NSA H
2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 2 for ll i, 1 i n. Then we sy tht S is n-generted semigroup. Generl references on numericl semigroups include [2, 6, 7, 8]. The Frobenius problem tkes its nme from the fct tht Frobenius is sid to hve mentioned it repetedly in his lectures [3]. However, the first published work on this problem ppers to be due to Sylvester [13] where he determined the number of elements of N \, b where nd b re reltively prime. Though not stted explicitly in [13] (or in the often cited [12]), it is suspected tht Sylvester knew tht g (, b ) = b b nd this fct is typiclly ttributed to him. Given such simple formul for the Frobenius number of two-generted semigroup, it is nturl to try to find the Frobenius number of n n-generted semigroup for other smll vlues of n. However, Curtis proved tht such closed-form expression cnnot be given for the Frobenius number of generl n-generted semigroup for n > 2 [4]. For this reson, Frobenius problem enthusists often consider semigroups whose genertors re of prticulr form. In [10], the uthors determine the Frobenius numbers of so-clled Fiboncci semigroups which re numericl semigroups of the form F i, F i+2, F i+k where F j denotes the j th Fiboncci number. In studying these semigroups, it is useful to recll the convolution property of Fiboncci numbers: F n = F m F n m+1 + F m 1 F n m for ll m, n Z + (where Z + denotes the set of positive integers). A Fiboncci semigroup is three-generted if nd only if 3 k < i (equivlently, < F i ). To see this, we consider two cses depending on the vlue of k. If k = i, then F i+k = F 2i = L i F i F i, F i+2 where L i denotes the i th Lucs number. If k > i, then nd so F i+k F i, F i+2. F i+k F 2i+1 = F i F i+2 + F i 1 F i+1 > g ( F i, F i+2 ) In this pper, we consider semigroups of the form S =, + b, 1 + b where > nd gcd(, b) = 1. Such semigroup will be clled generlized Fiboncci semigroup. Notice tht if = F i nd b = F i+1, then + b = F i+2 nd 1 + b = F i 1 + F i+1 = F i+k nd so every Fiboncci semigroup is generlized Fiboncci semigroup. Using method of S. M. Johnson [9], we find the dul of generlized Fiboncci semigroup. Recll tht the dul of numericl semigroup S is defined to be B(S) := {x N : x + (S \ {0) S. It is immedite tht g(s) B(S) for ny numericl semigroup S N s g(s) + s > g(s) for ll s Z +. Moreover, g(s) = mx {x B(S) : x / S.
3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 3 provided S N. Hence, we determine the Frobenius number of generlized Fiboncci semigroup obtining the result of [10] s corollry. This pper is orgnized s follows. Section 2 outlines Johnson s method nd pplies it to find the dul of generlized Fiboncci semigroup. Section 3 contins results relting the dul nd Lipmn semigroups. The pper concludes with Section 4 where severl open problems re posed. 2. Johnson s method We begin this section with review of S. M. Johnson s method [9] for determining the dul of semigroup generted by three reltively prime positive integers. Let S := 1, 2, 3 where 1, 2, nd 3 re pirwise reltively prime. Suppose N B(S) \ S. Then N + i S for i = 1, 2, 3; tht is, N = y ij j + y ik k i for some y ij, y ik N. Since the i re reltively prime, the semigroup generted by ny two of them hs Frobenius number. Hence, ny sufficiently lrge integer will be contined in such semigroup. Let L i := min {c : c i j, k. Then there exist x ij, x ik N such tht L i i := x ij j + x ik k. According to [9, Theorem 3], x ij nd x ik re positive integers nd re unique. Recll tht N = y y = y y Then nd so N = { (L 2 1) 2 + (x 13 1) 3 1 if y 21 < y 31 (x 21 1) 2 + (L 3 1) 3 1 if y 31 < y 21. { (L2 1) B(S) \ S = 2 + (x 13 1) 3 1, (x 21 1) 2 + (L 3 1) 3 1. Next, we pply this method to generlized Fiboncci semigroup. Consider S :=, + b, 1 + b where > nd the genertors of S re pirwise reltively prime. Note tht ( + b) = 2 + ( 1 + b)
4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 4 From this nd the rgument [9, p ], it follows tht In ddition, we find tht L 1 = ( + b) 2 L 2 =, x 21 = 2, nd x 23 = 1. x 12 = x 13 = L 2 = x 21 = 2 x 23 = 1 L 3 = + 1 As consequence, B(S) \ S = {( + b 2 Fk This proves the next two results. ) 2 b, ( 2 2) + ( 1 + b ) b. Proposition 1 Assume >. If S =, + b, 1 + b is generted by three pirwise reltively prime integers, then the dul of S is ) B(S) = S {( + b 2 2 b, ( 2 2) + ( 1 + b ) b. Fk Theorem 2 The Frobenius number of S =, + b, 1 + b where > nd the genertors of S re pirwise reltively prime is ) g(s) = mx {( + b 2 2 b, ( 2 2) + ( 1 + b ) b. Fk This theorem gives formul for g (F i, F i+2, F i+k ), due to Mrin et l. [10], when = F i nd b = F i+1. However, we should point out tht the technique used in [10], different from ours, llowed the uthors not only to stte explicitly when the mximum is obtined in ech cse but lso to give formul for the genus (mening N \ S ) of such Fiboncci semigroups. 3. Duls nd Lipmn semigroups In this section, we compre two chins of semigroups. One of the chins is bsed on the dul construction. The other chin rises by tking Lipmn semigroups. We first describe the Lipmn semigroup. Then we relte it to the dul of S. Finlly, we consider these for Fiboncci semigroups.
5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 5 Suppose S = 1, 2,..., n is numericl semigroup with 1 < < n. The Lipmn semigroup of S is defined s L(S) := 1, 2 1,..., n 1. Clerly, S L(S). Moreover, B(S) L(S) since x B(S) implies x + 1 = n i=1 c i i for some c i N. Given numericl semigroup S, both its dul B(S) nd its Lipmn semigroup L(S) re numericl semigroups. Hence, one my iterte the B nd L constructions to obtin two scending chins of numericl semigroups nd B 0 (S) := S B 1 (S) := B(B 0 (S)) B h+1 (S) := B(B h (S)) L 0 (S) := S L 1 (S) := L(L 0 (S)) L h+1 (S) := L(L h (S)) s in [2]. We will refer to these s the B- nd L-chins. Notice tht for S N, S B(S) nd S L(S). This together with the fct tht N \ S is finite implies tht there exist smllest non-negtive integers β(s) nd λ(s) such tht B β(s) (S) = N 0 = L λ(s) (S). Since nd B 0 (S) = S = L 0 (S), B 1 (S) L 1 (S), B β(s) (S) = N 0 = L λ(s) (S), it is nturl to compre the two chins. In [2] the uthors suggest tht B j (S) L j (S) for ll 0 j β(s). While true for two-generted semigroups, this continment my fil in generl; there re exmples of four-generted semigroups T for which B 2 (T ) L 2 (T ); see [5]. This prompts the question of whether or not B j (S) L j (S) for ll j 0 for three-generted semigroup S. Here, we consider this question for Fiboncci semigroups. Suppose tht S =, + b, 1 + b where = F i nd b = F i+1 ; tht is, suppose S = F i, F i+2, F i+k. Then the Lipmn semigroup of S is L 1 (S) =, b. Continuing this process yields the following result. Since < b, L 2 (S) =, b. Proposition 3 Given Fiboncci semigroup S = F i, F i+2, F i+k, L j (S) = F i j+1, F i j+2 for ll j, 1 j i 3. In prticulr, λ(s) = i 3.
6 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 6 Of course, one my obtin similr results for generlized Fiboncci semigroups. Becuse the description depends on sizes of nd b, we omit this here nd leve the detils to the reder. Since L j (S) is two-generted semigroup, L j (S) is symmetric for ll 1 j < λ(s). Therefore, L j (S) is mximl in the set of ll numericl semigroups with Frobenius number g (L j (S)). In prticulr, we notice tht if { F i = min x B Fi 1 Fi (S) : x 0, (1) F 2 2 k then ( g ) B Fi 1 Fi (S) = g(s) F 2 1 k ( F i Fi 1 ) 2 2 F i = g (L 1 (S)) [2, Proposition I.1.11]. Hence, if (1) holds nd then F i+1 B Fi 1 Fi (S), (2) F 2 1 k B Fi 1 Fi (S) = L F 2 1 1(S) k nd B j (S) L j (S) for ll nonnegtive integers j would follow from [5, Theorem 2.6]. Unfortuntely, becuse B 1 (S) is not three-generted, B 2 (S) my not be computed using the method in Section 2. Insted, one my compute this directly from the definition nd obtin the next result. Proposition 4 Given Fiboncci semigroup S = F i, F i+2, F i+k where nd B 2 (S) = Consequently, Fi, F i+2, F i+k, l F i+2, h F i+k if x 32 = 1, l F i if k > 4 or x 12 = 1, l F i+k if x 13 2 or x 13 = x 31 = 1, h F i if x 31 2 or x 31 = x 13 = 1, h F i+2 if x 12 2 or x 12 = x 21 = 1 h = ( 1) F i+2 + l = ( 1) F i+2 + ( ( ) ) Fi F i F i F i+k (( ) ) Fi + 1 F i 1 F i+k F i. B 2 (S) L 1 (S) L 2 (S). In light of Proposition 4, determining B j (S) for j > 3 will not be n immedite consequence of Johnson s method. We leve this (nd settling when (1) nd (2) hold) s problem for further study.
7 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 7 4. Conclusion In this pper, we determined the Frobenius number of generlized Fiboncci semigroups. In ddition, we lso obtined the dul of such semigroup. We leve s n open problems to 1. determine if B j (S) L j (S) for ech j 0 for Fiboncci semigroups S (or, more generlly, three-generted semigroups), nd 2. find the Frobenius number nd dul of other semigroups generted by generlized Fiboncci numbers. We conclude by mentioning nother interesting problem relting numericl semigroups nd Fiboncci numbers. M. Brs-Amoros conjectures tht the number of semigroups with prticulr genus g behves symptoticlly s the Fiboncci sequence [1]. References [1] M. Brs-Amoros, Fiboncci-like behvior of the number of numericl semigroups of given genus, Semigroup Forum, to pper. [2] V. Brucci, D. E. Dobbs nd M. Fontn, Mximlity properties in numericl semigroups nd pplictions to one-dimensionl nlyticlly irreducible locl domins, Memoirs Amer. Mth. Soc. 125/598 (1997). [3] A. Bruer, On problem of prtitions, Amer. J. Mth., 64 (1942), [4] F. Curtis, On formuls for the Frobenius numer of numericl semigroup, Mth Scnd. 67 (1990), [5] D. E. Dobbs nd G. L. Mtthews, On compring two chins of numericl semigroups nd detecting Arf semigroups, Semigroup Forum 63 (2001), [6] R. Fröberg, C. Gottlieb nd R. Häggkvist, On numericl semigroups, Semigroup Forum 35 (1987), no. 1, [7] R. Fröberg, C. Gottlieb nd R. Häggkvist, Semigroups, semigroup rings nd nlyticlly irreducible rings, Reports Dept. Mth. Univ. Stockholm no. 1 (1986). [8] R. Gilmer, Commuttive semigroup rings. Chicgo Lectures in Mthemtics. University of Chicgo Press, Chicgo, IL, [9] S. M. Johnson, A liner Diophntine problem, Cnd. J. Mth. 12 (1960),
8 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx 8 [10] J. M. Mrín, J. L. Rmírez Alfonsín nd M. P. Revuelt, On the Frobenius number of Fiboncci numericl semigroups, Integers 7 (2007), A14. [11] J. L. Rmírez Alfonsín, The Diophntine Frobenius Problem, Oxford Lecture Series in Mthemtics nd its Applictions 30, Oxford University Press, [12] J. J. Sylvester, Mthemticl questions with their solutions, Eductionl Times 41 (1884), 21. [13] J. J. Sylvester, On Subvrints, i.e. semi-invrints to binry quntics of n unlimited order, Amer. J. Mth. 5 (1882), no. 1-4,
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