ner 9 
XX. Note on the ivumbers of Bernoulli and Kuler, and a new 
Theorem concerning Prime Numbers. By J. J. SYLVESTER, 
M.A., F.R.S., Professor of Mathematics at the Royal Academy, 
Woolwich*. 
Be the accepted continental notation, I denote by 
Bt the positive value of the coefficient of 7?” in 
aon ge 
multiplied by the continual product 1.2.3...7. 
The law which governs the fractional part of B, was first 
given in Schumacher’s Nachrichten, by Thomas Clausen in 1840 ; 
ahd almost immediately afterwards a demonstration was freien 
by Professor Staudt in Crelle’s Journal, with a reclamation of 
priority, supported by a statement of his ‘having many years pre- 
viously communicated the theorem to Gauss. 
The law is this, that the positive or negative fractional residue 
of B, (according as n is odd or even) is ‘made up of the simple 
sum of the reciprocals of all the prime numbers which, respect- 
ively diminished by unity, are contained in 2n. The proof, which 
is of an inductive kind, is virtually as follows: Suppose the law 
holds good up to (n—1) ‘inclusive ; if we expand & (#)?” under the 
form — am, we shall evidently obtain aoe +B,, under the 
Ge 
form of a finite series, of which the terms are numerical mul- 
tiples of the products of powers of # by the Bernoullian num- 
bers of an order inferior to the nth. If, now, we make x equal 
to the product of all the primes which, diminished by unity, are 
contained in 2n, it will at once be seen (on inspection of the 
series) that all its terms become integer numbers, and con- 
27 
sequently at +B, becomes an integer; and therefore the law 
will hold good up to n, since it may easily be shown, by an 
application of Vermat’s theorem and elementary arithmetical 
considerations, that if N be the product of any prime numbers 
whatever, and if p is the general name of such . them as dimi- 
nished by unity are factors of w, then aoe 
EEO integer. 
Hence, since the law holds good forn= x , it is universally true. 
* Communicated by the Author. 
+ Were it not for the general usage being as stated in the text, I cer- 
tainly think it would be far more convenient to use a notation agreeing 
with the continental method as to sion, and nearly, but not quite, with 
Myr. DeMorgan’s as to quantity, viz. to understand by Bz the coefficient of 
et 
i” in it an taken positively, so that Bn should be equal to zero for all 
the odd values of n, not excepting n=1, 
