128 Prof. Sylvester on the Numbers of Bernoulli and Kuler, 
This theorem, then, of Staudt and Clausen, inter alia, gives a 
rule for determining what primes alone enter into the deno- 
minators of the Bernoullian numbers when expressed as frac- 
tions in their lowest terms; it enables us to affirm that only 
simple powers of primes enter into those denominators, and to 
know @ priort what those prime factors are. This note is 
intended to supply a law concerning the numerators of the Ber- 
noullian numbers, which I have not seen stated anywhere, and 
which admits of an instantaneous demonstration, to wit, that the 
whole of 2 will appear im the numerator of Bn, save and except 
such primes, or the powers of such primes, as we know by the 
Staudt-Clausen law must appear in the denominator. 
I am inclined to believe that this law of mine was not known, 
at all events, in 1840, from the circumstance that in Rothe’s 
Table, published by Ohm in Crelle’s Journal in that year, which 
gives the values of B, up to n=81, the numerators are, with one 
exception {about to be named), all exhibited in such a form as 
to show such low factors as readily offer themselves, but for By. 
the fact of the divisibility of the numerator by 23 is not in- 
dicated. This numerator is 596451111598912163277961, 
which in fact =23 x 25982657025822267968607. It is obvious, 
indeed, under my law, that whenever p is a prime number other 
than 2 and 8, the numerator of B, must contain p, because in 
such case p—1 cannot be a factor of 2p. When p=3 or p=2, 
2 p always contains (p—1), so that 2 and 8 are necessarily con- 
stant factors of the Bernoullian denominators, and can there- 
fore never appear in the numerators. In Schumacher the law of 
the denominator is given as “a passing” (or chance ?) ‘ speci- 
men” of a promised memoir by Clausen on the Bernoullian 
numbers, which I shall feel obliged if any of the readers of this 
Magazine will inform me whether anywhere, and if so, where it 
has appeared. Now for my demonstration of the law of the nu- 
merators. 
By definition, B,, =II(2n) x coefficient of ¢?”-! in a Let 
pw be any integer number; then + (u?”—1)B,=TII(2n) x coeffi- 
: bi at bb 
cient of ¢2"-! in eT ay == 
(—1) — (eH -DF 4 eH—DEL ,, +e?) 
evt— J ‘ 
——, or 1n 
ef —]? 
or in 
elu 2)t 1 Qelu—S)E4t + (w—2)eh+ (u—1) 
si eM—Di+4 eH—24 .., tet] ; 
But obviously, by Maclaurin’s theorem, the coefficient of ¢?"~1 in 
