REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS. 1. Introduction

Transcription

1 Version of 5/2/2003 To appear in Advanes in Applied Mathematis REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS MATTHIAS BECK AND SHELEMYAHU ZACKS Abstrat We study the Frobenius problem: given relatively prime positive integers a,, a d, find the largest value of t the Frobenius number ga,, a d suh that d k= m ka k = t has no solution in nonnegative integers m,, m d We introdue a method to ompute upper bounds for ga, a 2, a 3, whih seem to grow onsiderably slower than previously known bounds Our omputations are based on a formula for the restrited partition funtion, whih involves Dedekind- Rademaher sums, and the reiproity law for these sums Introdution Given positive integers a < a 2 < < a d with gda,, a d =, the linear Diophantine problem of Frobenius asks for the largest integer t for whih we annot find nonnegative integers m,, m d suh that t = m a m d a d We all this largest integer the Frobenius number ga,, a d ; its study was initiated in the 9th entury One fat whih makes this problem attrative is that it an be easily desribed, for example, in terms of oins of denominations a,, a d ; the Frobenius number is the largest amount of money whih annot be formed using these oins For d = 2, it is well known most probably at least sine Sylvester [2] that ga, a 2 = a a 2 a a 2 For d > 2, all attempts to find expliit formulas have proved elusive Two exellent survey papers on the Frobenius problem are [] and [] Our goal is to establish upper bounds for ga,, a d The literature on suh bounds is vast; it inludes results by Erdős and Graham [8] a 2 ga,, a d 2a d a, d Selmer [] ad 3 ga,, a d 2a d a d, d and Vitek [3] 4 ga,, a d 2 a 2 a d Mathematis Subjet Classifiation D04, 05A5, Y6 Key words and phrases The linear Diophantine problem of Frobenius, upper bounds, Dedekind-Rademaher sums, reiproity laws

2 2 MATTHIAS BECK AND SHELEMYAHU ZACKS Here a < a 2 < < a d, and x denotes the greatest integer not exeeding x Davison [6] established the lower bound 5 ga, a 2, a 3 3a a 2 a 3 a a 2 a 3 Experimental data [4] shows that Davison s bound is sharp in the sense that it is very often very lose to ga, a 2, a 3 On the other hand, the upper bounds given by 2, 3, and 4 seem to be quite large ompared to the atual Frobenius numbers In this paper, we derive a method of ahieving sharper upper bounds for the Frobenius number Our results are based on a formula for the restrited partition funtion Setion 2, whih involves Dedekind-Rademaher sums, and the reiproity law for these sums Setion 3 The main result is derived in Setion 4; omputations whih illustrate our new bounds an be found in Setion 5 We fous on the first non-trivial ase d = 3; any bound for this ase yields a general bound, as one an easily see that ga,, a d ga, a 2, a 3 if a, a 2, and a 3 are relatively prime If not then we an redue by one variable at a time: Again by the definition of the Frobenius number, ga,, a d ga,, a d if a,, a d are relatively prime If not, we an use a formula of Brauer and Shokely [5]: If n = gd a,, a d then 6 ga,, a d = n g Hene a n,, a d n, a d n a d a ga,, a d n g n,, a d n a d n 2 The restrited partition funtion We approah the Frobenius problem through the study of the restrited partition funtion { } p {a,,a d }n = # m,, m d Z d 0 : m a m d a d = n, the number of partitions of n using only a,, a d as parts In view of this funtion, the Frobenius number ga,, a d is the largest integer n suh that p {a,,a d }n = 0 In the d = 3 ase, we an additionally assume that a = a, b = a 2, and = a 3 are pairwise relatively prime, a simplifiation due to Johnson s formula [9]: if n = gda, b then a ga, b, = n g n, b n, n This identity is a speial ase of 6 In the ase that a, b, are pairwise relatively prime, Bek, Diaz, and Robins derived the following result for the partition funtion p {a,b,} [3, Theorem 3]: p {a,b,} n = n2 2ab n 2 ab a a b 2 b b a 7 ab S n b, ; a S n, a; b S n a, b;

3 REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS 3 Here [3, Equation 4] 8 S t a, b; = m=0 a bm t m, where aa mod and x = x x /2, is a speial ase of a Dedekind-Rademaher sum; we will disuss these sums in the next setion To bound the Frobenius number from above, we need to bound p {a,b,} from below, whose only nontrivial ingredients are the Dedekind-Rademaher sums A lassial bound for the Dedekind- Rademaher sum yielded in [3] the inequality ga, b, ab a b a b, 2 whih is of omparable size to the other upper bounds given by 2, 3, and 4 However, we will show that one an obtain bounds of smaller magnitude 3 Dedekind-Rademaher sums The Dedekind-Rademaher sum [0] is defined for a, b Z, x, y R as b ak y k y Ra, b; x, y = x, b b where k=0 x = { x if x Z, 0 if x Z Rademaher s sum generalizes the lassial Dedekind sum Ra, b; 0, 0 [7] An easy bound for the Dedekind-Rademaher sum Ra, b; x, 0 an be obtained through the Cauhy-Shwartz inequality: if a and b are relatively prime then b ak k Ra, b; x, 0 = b x b 9 = = k=0 b k=0 b k=0 b k=0 ak b x k b x 2 b k=0 2 b k= k 2 b k b 2 2 k b b 2 b b b 2 b 6b 2 4 6b

4 4 MATTHIAS BECK AND SHELEMYAHU ZACKS In the third and fourth step we use the periodiity of x An important property of Ra, b; x, y is Rademaher s reiproity law [0]: if a and b are relatively prime then 0 Ra, b; x, y Rb, a; y, x = Qa, b; x, y Here { Qa, b; x, y = 4 a 2 b ab a b if both x, y Z, x y a 2 b ψ 2y ab ψ 2ay bx b a ψ 2x otherwise, where ψ 2 x = x x 2 x x /6 denotes the periodi seond Bernoulli funtion Among other things, this reiproity law allows us to ompute Ra, b; x, y in polynomial time, by means of a Eulidean-type algorithm using the first two variables: simply note that we an replae a in Ra, b; x, y by the least residue of a modulo b To express S in terms of R, we rewrite 8 as S t a, b; = m=0 a bm t m { 4 if t, 2 a t 2 b t otherwise Aordingly, { S t a, b; = R a b, ; a t, 0 4 if t, 2 a t 2 b t otherwise To ease our omputations, we bound this as S t a, b; R a b, ; a t, Upper bounds for ga, b, To bound S t a, b; from below whih yields an upper bound for ga, b,, we use an interplay of 0 and 9 to obtain a bound for the Dedekind-Rademaher sum orresponding to S t, aording to The idea is to redue the arguments of the Dedekind-Rademaher sum after the appliation of 0, whih means that the bound given by 9 will be more aurate To illustrate this, let be the least nonnegative residue of a b modulo Then 2 R a b, ; a t, 0 = R, ; a t, 0 = Q If = then the right-hand side an be simplified, as R, ; a t, 0, ; 0, a t R, ; 0, a t = 0 If then the Dedekind-Rademaher sum on the right-hand side of 2 an be bounded via 9 sharper then the Dedekind-Rademaher sum on the left-hand side In fat, by a repeated appliation of 0, we an ahieve bounds whih are even better To keep the omputations somewhat simple, we apply 0 one more and illustrate what this proess yields in terms of lower bounds for S t Let 2 be

5 REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS 5 the least nonnegative residue of modulo If 2 = then R a b, ; a t, 0 = Q, ; a t, 0 = Q, ; a t 3, 0 = Q, ; a t, 0 as R, ; a t 4 R a b, ; a t, 0, 0 = 0 If 2 then 2 an be refined as = Q, ; a t, 0 = Q, ; a t, 0 R 2, ; 0, a t Q 2, ; 0, a t R, ; 0, a t R, ; 0, a t Q, ; 0, a t, R, 2 ; a t, 0 The Dedekind-Rademaher sum on the right-hand side an be bounded aording to 9 as 5 R, 2 ; a t, We still need to bound Q ψ 2 has a minimum of /2 at x = /2 and a maximum of /6 at x = 0 These extreme values yield for 4 2 if t, the lower bound 6 Q Q, ; a t, 0 = ψ 2 a t otherwise,, ; a t, = Q low,, 24 as well as the upper bound 7 Q 2, ; 0, a t These inequalities yield the following = Q up 2, Proposition Suppose a and b are relatively prime to Let be the least nonnegative residue of a b modulo, and let 2 be the least nonnegative residue of modulo i If = then S t a, b; ii If and 2 = then S t a, b; 2 iii If and 2 then S t a, b;

6 6 MATTHIAS BECK AND SHELEMYAHU ZACKS Proof i Use 2 with = in : S t a, b; R a b, ; a t, 0 2 = Q, ; a t, Here the last inequality follows from 6 ii Use 3 in together with the bounds 6 and 7: S t a, b; R a b, ; a t, 0 2 = Q, ; a t, 0 Q, ; 0, a t iii Use 4 with the bounds given in 5, 6, and 7: S t a, b; R a b, ; a t, 0 2 = Q, ; a t, 0 Q 2, ; 0, a t R , 2 ; a t, 0 2 These lower bounds an be ombined with 7 and the quadrati formula to give an upper bound on the Frobenius number Theorem Suppose a, b, and are pairwise relatively prime Denote the lower bounds for S t b, ; a, S t, a; b, and S t a, b; aording to the previous proposition by α, β, and γ, respetively Then ga, b, 4 a b 2 6 a2 b 2 2 2ab α β γ 2 a b One should note that α β γ is negative We an see that the growth behavior of this upper bound is dominated by 2ab α β γ under the square root This means that if we an make α β γ somewhat smaller than mina, b, then we get a bound whih grows onsiderably less that the bounds given by 2, 3, and 4 In fat, we an see this differene in example omputations already when we use the bounds α, β, γ as given by our proposition What is more important, however, is the fat that we an easily obtain even better bounds by improving our proposition through additional appliations of Rademaher s reiproity law 0 We illustrate this with the following algorithm, whose result is a bound on S t a, b;, whih an be used in the above theorem instead of the bounds oming from the proposition

7 REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS 7 Algorithm Input: a, b, pairwise relatively prime and N number of iterations Output: lower bound S for S t a, b; _ := - a^{-} b S := 0 n := modulo least nonnegative residue REPEAT { _2 := modulo _ least nonnegative residue S_ := S Q_low _, S_2 := S_ - Q_up _2,_ IF _ = THEN S := S_ Else S := S_2 IF _ = OR _2 = OR n = N THEN BREAK := _2 _ := _ n := n modulo _2 least nonnegative residue } IF _ > AND _2 > 2 THEN S := S - sqrt_2/2 /6 _2 - /4 _2/2 /6 _2 S := S - /2 The algorithm repeats the steps desribed in the proposition N times, at eah step bounding Q oming from Rademaher reiproity aording to 6 and 7 It stops prematurely if one of the variables is, in whih ase the remaining Dedekind-Rademaher sum is zero 5 Computations In the present setion we illustrate the newly proposed upper bound for ga, b, numerially In order to ompare the results also with the lower bound given by Davison 5 we present here the values fa, b, = ga, b, a b For these Frobenius numbers, Davison s lower bound is fa, b, 3 z, where z = ab In [4] we presented together with David Einstein an algorithm for the exat omputation of fa, b, Einstein omputed admissible see [4] values of fa, b, for relatively prime arguments hosen at random from the set {3,, 750} In [4] we arrived at the empirial onjeture that fa, b, ab 5/4 The objetives of our urrent presentations are:

8 8 MATTHIAS BECK AND SHELEMYAHU ZACKS i to ompare our new upper bound with the known upper bound, whih is, aording to 2, 3, and 4, min 2 a 3 a, 2b 3, 2 b 2 a b ; ii to ompare our new upper bound with the onjetured upper bound z 5/4 ; iii to ompare the new upper bound to the true value of fa, b, z Upper Bounds Figure The new and old upper bounds for the Frobenius number For these objetives we omputed the new upper bound and the known upper bound for two thousand values of a, b,, randomly hosen from Einstein s data In all omputations we used the minimum of the lower bounds given by the proposition and the algorithm for N = 2 in the theorem to obtain an upper bound for fa, b, In Figure we plot the new upper bound and the known upper bound as funtions of z Among the 2000 points only less than 00 have known upper bounds smaller than the new upper bound In 50% of the ases the ratio of the known upper bound to the new upper bound is greater than 244 In Figure 2 we plot the known upper bound as a funtion of the new upper bound This figure omplements the inferenes from Figure In Figure 3 we plot the new upper bound as a funtion of z, and ompare the points with Davison s lower bound 5 and with the onjetured upper bound z 5/4 We see that most values of the new

9 REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS 9 New Upper Bound Known Upper Bound Figure 2 The new and old upper bounds ompared upper bound are smaller than z 5/4 This gives additional redene to the empirial onjeture in [4] Finally, in Figure 4 we plot the points of the new upper bound versus the true fa, b, values We see that even for large values of fa, b, there are ases where the new upper bound yields lose values In 50% of the ases the ratio of the new upper bound to the true fa, b, is greater than 2 6 Final remarks As stated in the last setion, we used only two iterations N = 2 in our algorithm to ompute bounds for S t a, b,, whih in turn lead to bounds on the Frobenius number fa, b, It is interesting to ompare these values with the ones we get when just using the proposition, that is, one iteration N = Figure 5 illustrates this omparison It is reasonable to expet even better results when one uses more than two iterations in the algorithm However, we found that at least for our range of variables in the vast majority of ases the algorithm terminates prematurely after one or two iterations; aordingly, there is not muh gain from inreasing the number N of iterations

10 0 MATTHIAS BECK AND SHELEMYAHU ZACKS z New Upper Bound Figure 3 The new upper bounds and the onjeture The question whih remains, even with a higher number of iterations, is how our bound an possibly be improved The quality of the upper bound for fa, b, learly depends only on the quality of the lower bound for S t a, b,, and this bound is omputed in our algorithm There are three steps in the algorithm where we use bounds, namely, when adding[subtrating] Q low [Q up ], respetively, in the seond to last step where we use the Cauhy-Shwartz inequality, and in the last step where we use Let us assume that we use enough iterations so that we only leave the REPEAT loop when = or 2 = ; in this ase we don t use the Cauhy-Shwartz inequality The very last step does not entail a rude inequality, as we might be off maximally by one This leaves the bounds Q low and Q up given by 6 and 7, respetively, and this is where we lose quality: here we might be off by a frational fator of /, whih then gets multiplied in the theorem, with possibly grave onsequenes We lose with a remark on the general Frobenius problem, that is, with an arbitrary number of arguments a,, a d There are formulas analogous to 7 for d > 3 [3]; they involve higherdimensional analogs of Dedekind-Rademaher sums However, it is not lear how to ompute them effiiently Aording to [2], it is possible to ompute these ounting funtions quikly; we just don t know an atual way to do so It is our hope that the ideas in this paper an be extended to this general setting

11 REFINED UPPER BOUNDS FOR THE LINEAR DIOPHANTINE PROBLEM OF FROBENIUS F Numbers New Upper Bound Figure 4 The new upper bounds ompared to the Frobenius numbers Referenes J L Ramirez Alfonsin, The diophantine Frobenius problem, Report No 00893, Forshungsinstitut für diskrete Mathematik, Universität Bonn, Alexander I Barvinok, A polynomial time algorithm for ounting integral points in polyhedra when the dimension is fixed, Math Oper Res 9 994, Matthias Bek, Riardo Diaz, and Sinai Robins, The Frobenius problem, rational polytopes, and Fourier Dedekind sums, J Number Theory , 2 4 Matthias Bek, David Einstein, and Shelemyahu Zaks, Some experimental results on the Frobenius problem, Preprint arxiv:mathnt/ , to appear in Exp Math Alfred Brauer and James E Shokley, On a problem of Frobenius, J Reine Angew Math 2 962, MR 26 #63 6 J L Davison, On the linear Diophantine problem of Frobenius, J Number Theory , no 3, MR 95j:033 7 R Dedekind, Erläuterungen zu den Fragmenten xxviii, Colleted works of Bernhard Riemann, Dover Publ, New York, 953, pp P Erdős and R L Graham, On a linear diophantine problem of Frobenius, Ata Arith 2 972, MR 47 #27 9 S M Johnson, A linear diophantine problem, Canad J Math 2 960, MR 22 # H Rademaher, Some remarks on ertain generalized Dedekind sums, Ata Arith 9 964, MR 29 #72 Ernst S Selmer, On the linear Diophantine problem of Frobenius, J Reine Angew Math 293/ , 7 MR 56 #246 2 J J Sylvester, Mathematial questions with their solutions, Eduational Times 4 884, 7 78

12 2 MATTHIAS BECK AND SHELEMYAHU ZACKS New Upper Bound New Upper Bound Figure 5 The new upper bounds ompared iteration vs 2 iterations 3 Yehoshua Vitek, Bounds for a linear diophantine problem of Frobenius, J London Math So , MR 5 #028 Department of Mathematial Sienes, Binghamton University SUNY, Binghamton, NY , USA address: matthias@mathbinghamtonedu Department of Mathematial Sienes, Binghamton University SUNY, Binghamton, NY , USA address: shelly@mathbinghamtonedu