48 
Mr Babbage on a Theorem in Prime Number^. 
(S)=(-:) (p+(0(rfi)'+G) (*)+*■'• 
Let q—f — w, and this becomes, 
G) G) 
.p- 
-b I - 
All the terms except tlie first and last are divisible by and 
by p if both numbers are primes ; therefore we have, 
^ p-\-n.p-\-n — 1 + l jp.p — ^ 
1.2.3 p 1. 
always divisible by n p, when the numbers n p are primes : if 
li—n, this resolves into the former theorem. 
If p is a prime, and greater than n, then since ^ ^ o ^ 
is divisible by p, we have, 
p n. p 71 — 1 n + 1 
always divisible by p, whatever n may be, if p = a prime, other- 
wise, it is not divisible. 
This expression is also divisible by n, for the numerator is' 
91 1. -h 2. -f p ; and if this be arranged according to the 
powers of n, the term independent on n will, when divided by 
the denominator, leave unity, which is destroyed by the —1, 
and all the remaining terms are divisible by so that the ex- 
pression 
p-\-n.p-^n- — 1...714-1 
1.2 p 
is abvays divisible by p w, if p is a prime, otherwise it is not di- 
visible by p, but only by n. 
By considering the coefficients of the cube, and other 
powers of 
1-h 
(t) + (t) + (I) +’ 
we might arrive at other theorems respecting prime numbers ; 
but the number of the combinations which occur in all the 
higher powers, seem to exclude tlmt simplicity in the expression^ 
