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Transcript
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
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DEPARTMENT OF MATHEMATICS
UNIVERSITY OF SOUTH CAROLINA AT SUMTER
1 LOUISE CIRCLE
SUMTER, SC 29150
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are completed.
1998]
281