and a new Theorem concerning Prime Numbers. 135 
according as 2n does not or does contain (w—1), which proves 
the law for the numerators; and so if pw‘ is a factor of n, but 
(u—1) not a factor of 2n, x will vanish, and 2”—1 will not con- 
tain “; hence (u?”—1)B,, and consequently B,, will be the pro- 
duct of «* by an integer relative to ~, which proves my nume- 
rator law. 
So by extending the same method to the generating function 
ef + 
Q4 Q2n 
secO=H,+E + Baya ga te t En pa gay t he 
6? 
IT, 2° +E 
every prime number yp of the form 4n+1, such that (u—1) isa 
factor of 2n, will be contained in H,,; and every such factor, when 
p is of the form 4m—1], will be contained in H,+(—)"2. 
I call the numbers E,, E,,... HE, EHuler’s Ist, 2nd,...nth 
numbers, as Kuler was apparently the first to brmg them into 
notice. In the Institutiones Caleuli Diff. he has calculated their 
values up to H, inclusive: in this last there isan error, which is 
specified by Rothe in Ohm’s paper above referred to; had Kuler 
been possessed of my law this mistake could not have occurred, 
as we know that H,+2 ought to contain the factors 19 and 7, 
neither of which will be found to be such factors if we adopt 
Kuler’s value of Ky, but both will be such if we accept Rothe’s 
corrected value. But in still following out the same method, I 
have been led, through the study of Bernoull’s and the allied 
numbers, and with the express aid of the former, to a perfectly 
general theorem concerning prime numbers, in which Bernoulli’s 
numbers no longer take any part. Fermat’s theorem teaches us 
the residue of g“~! in respect to w, viz. that it is unity; but I 
am not aware of any theorem being in existence which teaches 
, : #— lo] aia 
anything concerning the relation of Fe Be ty (or, which is 
the same thing, of the relation of gu—-1 to the modulus ?). I have 
obtained remarkable results relative to the above quotient, which 
[I will state for the simplest case only, viz. that where q¢ as well 
as # is a prime number. I find that when q is-any odd prime, 
Lo a 
be w—l 
where ¢,, Co, Cs). --Cy—1 are continually recurring cycles of the 
numbers 1, 2, 3,...7, the cycle beginning with that number 7! 
Co C3 Cu—1 
Sy se ae aes ks 3 
