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Note New Modular Properties of Bell Numbers JOURNAL OF COMBINATORIAL THEORY, Series A 58, 147-152 (1991) Note New Modular Properties of Bell Numbers NABIL KAHALE* Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Communicated by the Managing Editors Received December 4, 1989; revised March 23, 1990 Given a prime p, we exhibit an integer c such that the Bell numbers 3 Gil, B,+z> ...> B,+,-, are divisible by p. We calculate the residue of B, modulo P. 0 1991 Academic Press, Inc. 1. INTRODUCTION The Bell number B, is the number of partitions of a finite set whose cardinality is n. The modular properties of Bell numbers have been studied by many authors. It is well known that the sequence of the residues of Bell numbers modulo any integer satisfies a linear recurrence and is periodic [ 1, 68, 13, 143. When the modulus is prime, this recurrence is particularly simple. Touchard [16] shows the congruence where p is any prime number. The sequence of residues of B, modulo p is periodic [2, 173 and its period divides T= (pp - 1 )/(p - 1). Radoux [15] conjectures that T is the minimal period of this sequence for any prime p. He shows [14} that if the period of the residues is equal ,to T for a given prime p, then there exists a number c, depending on p, such that B, + 4 SE0 (mod p) for 1~ 4 < p. However, he did not give any general value for c. Radoux’s conjecture has been verified [12] for all primes p less than or equal to 17 but, to our knowledge, it has not been proven in the general case. In Section 3 we show, without assuming any unproven results, the following: * This research was supported by the Defense Advanced Research Projects Agency under Contract NOOO14-89-J-1988. 147 0097.3165/91 $3.00 Copyright 0 1991 by Academic Press, Inc. All rights of reproduction in any form reserved. 148 NABIL KAHALE THEOREM. Let c = (pJ’ - T)/(p - 1). Then c is a positive integer and B,,+,rO(modp) for l<q<p-1. (2) Furthermore, if p > 2, then B~,(_1)(P-1)(p-3)/8p-L T! (mod p). (3) In Section 4, we explore further modular properties of the Bell numbers and show the existence of a sequence of p consecutive Bell numbers having the same residue modulo p. 2. PRELIMINARIES Throughout this note, p refers to a prime number and 6 is the Kronecker symbol. If a is an element of a ring whose unit element is e, then a” stands for a(a-e)...(a-(n-1)e) and a* for a(u+e)~~~(u+(n--1)e). We say that two infinite integer sequences {u,} and {v,} are congruent modulo p if and only if for all n 3 0, the elements u, and v, have the same residue modulo p. Unless stated otherwise, all congruences are modulo p. From (1) and the recurrence relation B,, 1 =Ci (1) Bi it is not hard to show [16] that, for all IZ 2 0, B n+kp= Bn+i, (4) Bkp--k+l> (5) B n+ps=@,+Bn+i. (6) Other properties of the Bell numbers can be found in [3]. De Bruijn [4, p. 1041 gives their asymptotic behavior. The Stirling number of the second kind, denoted {i}, is defined as the number of partitions of the set { 1, . n} into k blocks. We have The reader is referred to [11] for other properties of the Stirling numbers. MODULAR PROPERTIES OF BELL NUMBERS 149 3. THE PROOF OF THE THEOREM Before proving the theorem, we prove the following lemma. LEMMA. For all n, q > 0, B n+y- (91 ,’ Proof of lemma. By combining (5) and (6), we have B,, + pq= qB, + B,,+ I (mod p). Using (lo), we prove by induction on k that where E is the shift operator that maps an infinite sequence (un> to the sequence {u, + r >. For the basis, (11) holds for k = 0. Now, assuming that it holds for k, the relation Ek+’ = (E- k) Ek implies that Ea( (B, > ) is congruent to {B,, + , jpk - kB,k} which is congruent to (B,@+I} by (10). Thus, (11) holds for k+ 1. Applying both sides of the identity Eq = xi (r} E’ to (B,) and using (11) completes the proof of the lemma. 1 We are now ready to prove the theorem. Proof of theorem. Since p is congruent to 1, modulo (p - 1 ), so is any power of p. Hence, pJ’ is congruent to 1, modulo (p - I), and so is T = pP-l + . + p + 1. Thus pp - T, which is obviously positive, is a multiple of p - 1, and so c is a positive integer. Since pp is congruent to 1, modulo T, so is (p - 1) c. Thus, pc=c+ 1 (mod T). Multiplying both sides of (12) by pq yields CPq+lscpQ+pq (mod T). Since the period of the residues of B, divides T, (13) shows that BCpu+,z B cp~+pq(mod p). But, using (lo), we have B,,,., = qB+ + B,,+I (mod p). By comparing these two congruences, we see that qB,, is a multiple of p. Since p is prime and q is not a multiple of p, we con&de that BcPqis a mu& tiple of p. An application of the lemma yields the first desired result (2). 150 NABIL KAHALE Now, we prove (3). Let A, be the p x p matrix 4, &,+I... Bn+p-I By (1) we have detA,+,=detA,, and thus, det A, E det A (mod p), (14) where A = A, is the Hankel matrix built on the Bell numbers. But, u! .ng (1) and (2), we see that Br+k= 6 kpB, for 1 <k < 2p - 2. In other words, B, 0 ... 0 0 . 00:’ ; 0 B, 0 ... 0 Hence, det A, = (- l)(p-1u2 B,P. C ombining this relation with (14) and using Fermat’s theorem yields (15) If i lies between 0 and p - 1, then, by [lo, p. 681, we have (P; ‘) - (- 1)’ and therefore i! (p - 1 - i)! E (- l)i (p - l)!. Since, by Wilson’s theorem, (p-1)!r-1,wehavei!(p-1-i)!~(-1)i+1.By[5],however,detA= I-&‘:: k!, and so detA=p~!‘p~y2i!(p-~-~)!~(-l)‘~-~”p+~’,6P~!. i=O 2 We complete the proof of (3) by replacing det A by its value in (15). g Remarks. Hardy and Wright [9, p. 881 show that p= 4m + 3 implies ((p - 1)/2)! = ( - 1 )‘, where v is the number of quadratic non-residues less than p/2. In this case, we have B, = ( - l)mf”. If p = 4m + 1, then ((p - 1)/2)! is a square root of - 1, and so B, is also a square root of - 1. MODULARPROPERTIESOFBELLNUMBERS 151 4. OTHER RESULTS For O<k<:p-2, let c,=c+l-pk and ~,~~=ci-l-~~~‘+T. We leave, as an exercise to the reader, the proofs of the following congruences, which hold for 0 < IZ < p: Bc+~+y=Bc+~+q,m (161 B,+,-(q-k)‘%, (171 BckPnE (- l)n k”B,., (181 B,+l-,=O. (191 Congruence (19) holds if s is of the form s = xi=, p”‘, where I> 1 and the nj have distinct residues modulo p. Applying (17) in the particular case k = p - 1 and q = 0 shows us that B cp-,, B+,+l> . ..> B+,+p-1 have the same residue modulo p as B,. When p= 5, we have T= 781, c= 586, c,=743, B,,, zz B,,,- Bsg9z BTg, G 0 (mod 5), and B,,, E B,,, z B744 E . - B,,, = 3 (mod 5). ACKNOWLEDGMENTS The author expresses his gratitude to Thomas Cormen, Ira Gessel, and Richard Stanley for their helpful comments and suggestions. REFERENCES 1. D. BARSKY, Analyse p-adique et nombres de Bell, C.R. Acad. Sci. Paris SPr. A 282 (1976), 1257-1259. 2. H. W. BECKER AND J. RIORDAN, The arithmetic of Bell and Stirling numbers, Amer. J. Math. 70 (1948), 385-394. 3. L. COMTET, “Advanced Combinatorics,” Reidel, Dordrecht, 1974. 4. N. G. DE BRUIJN, “Asymptotic Methods in Analysis,” Dover, New York, 1981. 5. P. DELSARTE, Nombres de Bell et polynbmes de Charlier, C.R. Acad. Sci. Paris Sk. A 287 (1978), 271-273. 6. P. FLAJOLET, On congruences and continued fractions for some classical combinatorial quantities, Discrete Math. 41 (1982), 145-153. 7. I. M. GESSEL, Congruences for Bell and tangent numbers, Fibonacci Quart. 11 (1981), 137-144. 8. I. M. GESSEL, Combinatorial proofs of congruences, in “Enumeration and Design,” pp. 157-197, Academic Press, Toronto, 1984. 9. G. H. HARDY AND E. M. WRIGHT, “An Introduction to the Theory of Numbers,” Oxford Univ. Press (Clarendon), Oxford, 1979. 10. D. E. KNUTH, “The Art of Computer Programming. Vol. 1. Fundamental Algorithms,” Addison-Wesley, Reading, MA, 1973. 582af58’1.II 152 NABIL KAHALE 11. R. L. GRAHAM, D. E. KNUTH, 0. PATASHNIK, “Concrete Mathematics; A Foundation for Computer Science,” Addison-Wesley, Reading, MA, 1989. 12. J. LEVINE AND R. E. DALTON, Minimum periods, modulo p, of first order Bell exponential integers, Math. Camp. 16 (1962), 41&423. 13. W. F. LUNNON, P. A. B. PLEASANTS,AND N. M. STEPHENS,Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith. 35 (1979), 1-16. 14. C. RADOUX, Nombres de Bell, module p premier, et extensions de degre p de Fp, CR. Acad. Sci. Paris Sir. A 281, 879-882. 15. C. RADOUX, Arithmttique des nombres de Bell et analyse p-adique, Bull.
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