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On Dibag's generalisation of von Staudt's theorem
JOURNAL
OF ALGEBRA
On Dibag’s
141, 42&421
(1991)
Generalisation
of Von Staudt’s
FRANCIS
Theorem
CLARKE
Department
of Mathematics and Computer Science, Universily College Swansea, Singleton Park, Swansea SA2 8PP, Wales* Communicated Received
by Walter
December
Feit
5. 1989
We show how a stronger version of a generalisation of von Staudt’s to I. Dibag (An analogue of the von Staudt-Clausen theorem, J. (1989), 519-523) follows from the “universal von Staudt theorems” Trans. Amer. Math. Sot. (The universal von Staudt theorems, 591-603).
0
1991 Academx
theorem due Algebra 125 of F.Clarke 315 (1989),
Press, Inc.
Given a set of primes S, Dibag [S] denotes by Q, the set of positive integers whose prime factors belong to S and defines the rational numbers b, by the identity x/L;’ (x) = C,“=, b,(x”/n!), where the series L,(x) is given by L,(x)=
c (-I)“-‘$. nsOs
In the notation of [2] the number b, can be obtained from the universal Bernoulli number fi,, by letting if i+loQ,, otherwise. Generalising the classical result of Clausen [3] and von Staudt [6] and the results of his earlier paper [4], Dibag proves in [ 5, Theorem 1.143 that if n is even then b, is congruent modulo Z to - C l/p, summing over those primes p E S such that p - 1 divides n. This result follows immediately from the congruence for B,, modulo Z[c, , ca, ...] given in [2, Corollary 63; indeed it follows from 11, Theorem 2). We show below how stronger con* (mafred
(3 uk. ac. swan. pyr).
420 0021-8693/91
$3.00
Copyright 0 1991 by Academic Press, Inc All rights of reproducrion in any form reserved
GENERALISATION
OF VON
STAUDT’S
THEOREM
421
gruences for the b, may be deduced from the results of [2], including results when n is odd. Dibag’s theorem may be written in the form pb, z - 1 mod pZ,,,, if p E S and p- 1 divides n, where Z,pj denotes the ring of p-local integers. Let v,(n) denote the exponent of p in n. THEOREM 1. Suppose that n is even and p is a prime such that p - 1 divides n. If p E S then
pb,=p-
1
mod p 1+ QAnQ(PI’
otherwise
mod p”pP(‘)ZCp,.
b, z 0 Proof:
Apply [2, Corollary
71.
If 2 4 S then b, = 0 for n odd. The same is true for odd n > 1 in the case where S is the set of all primes, for then 6, is the Bernoulli number B,. However, in general 6, will be non-zero for odd n. It follows immediately from [2, Theorem 51 that if n is odd then 6, E nZ.
REFERENCES
1. L. CARLITZ, The coefficients of the reciprocal of a series, Duke Math. J. 8 (1941), 689-700. 2. F. CLARKE, The universal von Staudt theorems, Trans. Amer. Math. Sot. 315 (1989), 591-603. 3. T. CLAUSEN, Theorem, Astronom. Nuchr. 17 (1840), 351-352. 4. I. DIBAG, An analogue of the von Staudt-Clausen theorem, J. Algebra 87 (1984), 332-341. 5. I. DIBAG, Generalization of the von StaudtClausen theorem, J. Algebra 125 (1989), 519-523. 6. K. G. C. VON STAUDT, Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend, J. Reine Anger. Math. 21 (1840), 373-374.
OF ALGEBRA
On Dibag’s
141, 42&421
(1991)
Generalisation
of Von Staudt’s
FRANCIS
Theorem
CLARKE
Department
of Mathematics and Computer Science, Universily College Swansea, Singleton Park, Swansea SA2 8PP, Wales* Communicated Received
by Walter
December
Feit
5. 1989
We show how a stronger version of a generalisation of von Staudt’s to I. Dibag (An analogue of the von Staudt-Clausen theorem, J. (1989), 519-523) follows from the “universal von Staudt theorems” Trans. Amer. Math. Sot. (The universal von Staudt theorems, 591-603).
0
1991 Academx
theorem due Algebra 125 of F.Clarke 315 (1989),
Press, Inc.
Given a set of primes S, Dibag [S] denotes by Q, the set of positive integers whose prime factors belong to S and defines the rational numbers b, by the identity x/L;’ (x) = C,“=, b,(x”/n!), where the series L,(x) is given by L,(x)=
c (-I)“-‘$. nsOs
In the notation of [2] the number b, can be obtained from the universal Bernoulli number fi,, by letting if i+loQ,, otherwise. Generalising the classical result of Clausen [3] and von Staudt [6] and the results of his earlier paper [4], Dibag proves in [ 5, Theorem 1.143 that if n is even then b, is congruent modulo Z to - C l/p, summing over those primes p E S such that p - 1 divides n. This result follows immediately from the congruence for B,, modulo Z[c, , ca, ...] given in [2, Corollary 63; indeed it follows from 11, Theorem 2). We show below how stronger con* (mafred
(3 uk. ac. swan. pyr).
420 0021-8693/91
$3.00
Copyright 0 1991 by Academic Press, Inc All rights of reproducrion in any form reserved
GENERALISATION
OF VON
STAUDT’S
THEOREM
421
gruences for the b, may be deduced from the results of [2], including results when n is odd. Dibag’s theorem may be written in the form pb, z - 1 mod pZ,,,, if p E S and p- 1 divides n, where Z,pj denotes the ring of p-local integers. Let v,(n) denote the exponent of p in n. THEOREM 1. Suppose that n is even and p is a prime such that p - 1 divides n. If p E S then
pb,=p-
1
mod p 1+ QAnQ(PI’
otherwise
mod p”pP(‘)ZCp,.
b, z 0 Proof:
Apply [2, Corollary
71.
If 2 4 S then b, = 0 for n odd. The same is true for odd n > 1 in the case where S is the set of all primes, for then 6, is the Bernoulli number B,. However, in general 6, will be non-zero for odd n. It follows immediately from [2, Theorem 51 that if n is odd then 6, E nZ.
REFERENCES
1. L. CARLITZ, The coefficients of the reciprocal of a series, Duke Math. J. 8 (1941), 689-700. 2. F. CLARKE, The universal von Staudt theorems, Trans. Amer. Math. Sot. 315 (1989), 591-603. 3. T. CLAUSEN, Theorem, Astronom. Nuchr. 17 (1840), 351-352. 4. I. DIBAG, An analogue of the von Staudt-Clausen theorem, J. Algebra 87 (1984), 332-341. 5. I. DIBAG, Generalization of the von StaudtClausen theorem, J. Algebra 125 (1989), 519-523. 6. K. G. C. VON STAUDT, Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffend, J. Reine Anger. Math. 21 (1840), 373-374.