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CONGRUENCES MOD pn FOR THE BERNOULLI NUMBERS A. Simalarides 196 Kifissias Str., Kifissia 14562, Athens, Greece {Submitted September 1996) 1. INTRODUCTION Let/? be a prime. In 1889 Voronoi proved the congruence 2 1 ( « - « ' • • *2k ) % - £, , sa L *- (modp), (1) P where £, a are positive integers such thatp does not divide a and p-\ does not divide 2k; B2k is the 2£th Bernoulli number. More general versions of this congruence can be found in [6] or [3]. Following Wagstaff, denote congruence (1) also by the symbol {a}. Adding together congruences {2}, {3}, and -{4}, we obtain the congruence •5=1 L. {2}+ {3}-{4} which, after some obvious cancellations in the right member, takes the form (2P-2k+3P-2k_4P-2k_l)B2L^ jy*-i 4k (mod/?)? (2) p/4<s<p/3 provided that p > 4. Several such identities are also obtainable in a way analogous to that shown above by using suitable variations of parameter a. Several authors used formulas of this type to test regularity via computer. The best result in this direction is the following one, due to Tanner and Wagstaff [5], which is valid for all primes p > 10, -2* (2p- + gP-2k -io?-2k- -1)^*- - ( l + 22fe-1+32fc-1 + 42k~ ) 4k + (1 + 22k-l+32k- 1 i +4 2fe " +i2 M 1 2> yyk- 1 -) no<s<9 -32*-1 -(2 2 *- 1 + 62k\T> -2 2 *- 1 60 <S< 2k-l -(2 2 *- l+42k -l+l22k-1) -1 -4£ 10 -(22M+42*-1) (3) 18 %s2k-1 120 ^ ^ (mod/?). 5 In formula (3), the sums in the right member contain a total of about p/lS terms [formula (2) contains about p112 terms while formula (1) contains (p -1) / 2 terms for a = 2]. All the applications of these formulas concerning Fermat's Last Theorem are now mainly of historical interest 276 [JUNE-JULY CONGRUENCES MOD pn FOR THE BERNOULLI NUMBERS after Wiles's proof [8] of FLT. There are congruences of various types for the Bernoulli numbers. Recent results on congruences for Bernoulli numbers of higher order can be found in [2]. We shall prove the following analog of formula (1). Theorem 1: Let ^ be a primitive Dirichlet character with modulus m > 2. If a > 2 is an integer such that m does not divide a, then f0 if % is even, m-\ sa (4) z0): m s=\ * ( 2 ) - 2 s=l where the bar means complex conjugation. The proof of Theorem 1 will be given in Section 2. Formula (4) can be written, equivalently, in the form fO if^iseven, m-\ sa X(s)(5) Z,SX(S) if^isodd, m s=\ m s=l L because of the formula s m-l s s m [mil] Y x( ) = z(2)~2; Y,Zis), s=l (6) which is valid for an odd primitive character % • We use formula (5) to obtain /^-divisibility criteria for Bernoulli numbers of the form n (2k-\)Pn+v * - M , ~ ; 2 * Criteria of this type are still of interest because of their connection with the invariants of the irregular class group of a properly irregular cyclotomic field [7] (cf. also [4], p. 189). Assume now that m = p , an odd prime. Let y/ be the character defined as the/?-adic limit y/(s) - lim sp for every s prime top. All the values of y/ belong to Z p , the ring of/7-adic integers. Moreover, y/(s) = spn~l (mod// 7 ), n>\. For an odd character, we have x - W2k~l? f ° r s o m e k>\, and ^ ) ^ - ^ X{S) - 5-^" ^ - sT^y-^Vk-l) ,(2 1) (mod//*), s sp-\P-2k) ( m o d pny Theorem 2: Let/? be a prime >3. If a is an integer such that/? does not divide a, then p-l [a-a p»-\p-2k) ]Bt(2Jt-l)/7"+l -I sa Pk~l)pn~l (mod// 1 ), (7) for every k > 1 such that p - l does not divide 2k. 1998] 277 CONGRUENCES MOD p n FOR THE BERNOULLI NUMBERS Proof: We consider the 17th Bernoulli polynomial y=ov Then, for the odd character % = y2^"1, w e have p-1 P k B l (v)-B y sr(s) = y p -w+ = w-w+v^' "yk-w+i S S //Z> Since p-\ (2k-l)p»+2 [(2k-l)p» + l](2k-l)p» (2Jt-l)/7"+l ^ 3 31 r U (2k-\)pn-\ ^ does not divide 2k, we obtain the congruence P-\ (mod TI«(')-W« ^ which, together with Theorem 1 and relation (5), yields the sought result. For n = 1, congruence (7) reduces to congruence (1) since V - ^ , = [(2t-l)p + 1 ] J ^ ± _ . f (mod,) because of Kummer's congruence. We can prove, using exactly analogous techniques and starting from (7), a pn-analog of congruence (3). Because of the obvious analogy between the proofs, the sought result follows simply by replacing expressions of the form Qp-2k Q2k-l ^-1 B2L 9 2k in congruence (3) with the respective expressions n np U 1 n(2k-l)p">" -\p-2k) J2Jc-l)p"-1 >A ' R n {2k-l)pn+V The following theorem then follows. Theorem3: Let/? be an odd prime >10, k>l, p-l n 1 2{p-2k)p - +9(p-2k)p"-1 _lQ(p-2k)pn-1 _l (2k-l)pn+l 2 = [1+2{2k~l)p ."-1 does not divide 2kand n > 1. Then /-Ot 1\„"-1 + 2{2k~l)p + 42k~l)p ] y sw-i)pn~l 10^^120 + [1 + 2(2k~1)p"~l + 3^2A:-1>^"1 + 4(2*-1)/?w~1 + ^C 2 ^- 1 )/ 7 "" 1 ] V s( 2k l n l - )p ~ 120^^9 _ 3(2k-l)p"- 1 y pk-l)jT 9 ^ ^ 30 278 l 1 _ |-2(2A:-l)/7"- + 6(2k-l)p"- 1 j y 18 <s< 1 s(2k-l)p"- 60 [JUNE-JULY CONGRUENCES MOD p" FOR THE BERNOULLI NUMBERS _ 2(2k-i)P"-1 y 1 s(2k-i)P"- __ r2(2k-i)P"-1 1 + 4(2k-i)P"- 1 + l2(2k-i)P"- 60 ^ A ^ 1 0 -J y pk-\)Pn 1 8 ^ " ^ 120 120 ^ A < 5 The congruence contains in the right member pi 18 terms only. 2, PROOF OF THEOREM 1 At first, we note that, obviously, - I -m* * ) = ! I>(4 5=1 L For integerj, 0<j<a, 7=1 (8) 5=0 define y 0(x) = 1 0 ifx = 0or2nj/a, if0<x<27rj/a, if 2;rj I a<x<2n, and continue O(JC) periodically with period 2;r over the real numbers. The function O(x) has the Fourier expansion where c = — f < D ( x K " ^ = — ( * " -1) 2K First, we assume that a <m. Then [//w/a] jm/a\ m-l m-l 5=0 5=1 / n __ 2/nr//i 5=1 2/rij n=—<x> Inijn r(x)i f 2n ±L (e~-l)x(n) n where 5=1 1998] 279 CONGRUENCES MOD p" FOR THE BERNOULLI NUMBERS As a consequence, y=i 2/r „ ~ 5=0 n=—co _ia> v •a n v-i Xto*--** <X)i" f 2^ i i n £ In XM Jr.L " Since « imjn (a 6 P ' [0 if n = 0(moda\ if«#0(moda), it follows that /=1 In „ ~ 5=0 na ^ «=-oo W = 4 ^ a ( a ) - a ) i m For even %, the last infinite sum is equal to zero while, for odd x, it is equal to 2Z(1, ~x). In view of the formula (cf [1], p. 336) L{\X) = m (2-z(2))r(x) [mil] I,Zis) % and relation (8), it follows that , [m/2] m-\ (9) ^ILWJ for a < m. It remains to prove the theorem for a > m. Then a = al+mt, where ax and t are integers and 0 < ax < m. Also m does not divide av We have m-1 m-ir 5ax +sf zO) 2Hk<*)=S Z 5=1 5=1 m-l SON m-l = S|"^Lk(ff) + r Z^( 5 )5=1 L ' " J 5=1 The last expression is zero for even % For odd % w e have, in view of (6) and (9), which proves the theorem fora>m. 280 [JUNE-JULY CONGRUENCES MOD pn FOR THE BERNOULLI NUMBERS REFERENCES 1. Z. I. Borevich & I. R. Shafarevich. Number Theory. New York: Academic Press, 1966. 2. F.T.Howard. "Congruences and Recurrences for Bernoulli Numbers of Higher Order." The Fibonacci Quarterly 32.4 (1994):316-28. 3. K. Ireland & M. Rosen. A Classical Introduction to Modern Number Theory. New York and Berlin: Springer-Verlag, 1982. 4. P. Ribenboim. 13 Lectures on Fermafs Last Theorem. New York and Berlin: SpringerVerlag, 1979. 5. J. W. Tanner & S. S. Wagstaff. "New Congruences for the Bernoulli Numbers." Math, of Comp. 48 (1987):341-50. 6. J. V. Uspensky & M. A. Heaslet. Elementary Number Theory. New York: McGraw-Hill, 1939. 7. H. S. Vandiver. "On the Composition of the Group of Ideal Classes in a Properly Irregular Cyclotomic Field." Monatsh f. Math u. Phys. 48 (1939):369-80. 8. A. Wiles. "Modular Elliptic Curves and Fermat's Last Theorem." Annals of Math. 142 (1995):443-551. AMS Classification Number: 11B68 Author and Title Index The AUTHOR, TITLE, KEY-WORD, ELEMENTARY PROBLEMS, and ADVANCED PROBLEMS indices for the first 30 volumes of The Fibonacci Quarterly have been completed by Dr. Charles K. Cook. Publication of the completed indices is on a 3.5-inch, high density disk. The price for a copyrighted version of the disk will be $40.00 plus postage for nonsubscribers, while subscribers to The Fibonacci Quarterly need only pay $20.00 plus postage. For additional information, or to order a disk copy of the indices, write to: PROFESSOR CHARLES K. COOK DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA AT SUMTER 1 LOUISE CIRCLE SUMTER, SC 29150 The indices have been compiled using WORDPERFECT. Should you wish to order a copy of the indices for another wordprocessor or for a non-compatible IBM machine, please explain your situation to Dr. Cook when you place your order aind he will try to accommodate you. DO NOT SEND PAYMENT WITH YOUR ORDER. You will be billed for the indices and postage by Dr. Cook when he sends you the disk. A star is used in the indices to indicate unsolved problems. Furthermore, Dr. Cook is working on a SUBJECT index and will also be classifying all articles by use of the AMS Classification Scheme. Those who purchase the indices will be given one free update of all indices when the SUBJECT index and the AMS Classification of all articles published in The Fibonacci Quarterly are completed. 1998] 281