< 46 ) 
Art. Vll.->-^Demonstratlon qf a Theorem relating to Prime 
N'umbers. By Charles Babbage, Esq. F. B. S. &c. Com- 
municated by the Author. 
The singular theorem of Wilson respecting Prime Numbers, 
which was first published by Waring in his Meditaiiones Aria^ 
J/yticoSy and to which neither himself nor its author could supply 
the demonstration, excited the attention of the most celebrated 
analysts of the continent, and to the labours of Lagrange and 
Euler we are indebted for several modes of proof; and more re- 
cently it has been considerably extended by the profound inves- 
tigations contained in the Disquisitiones Ariihmeticoe. 
It is well known that, in the theorem in question, a certain 
expression is asserted to be divisible by w, whenever that num- 
ber is a 'prime, but it is not divisible if n is not prime. In at- 
tempting to discover some analogous expression which should 
be divisible by whenever is a prime, but not divisible if n 
is a composite number, I met with those properties of primes 
which form the subject of the present paper. 
The theorem of Wilson asserts that 
1.2.S....?z — 1-f 1 
is always divisible by n when is a prime number, otherwise it 
is not. The theorem which I have arrived at is as follows^ 
that 
n -p \ .n-\- 2.7^ -|- 3 . . . . . . 2 71-— 1 
1.^.3 
n 
1 
— 1 
is always divisible by when n is a prime number, otherwise 
it is not. The demonstration is very simple. Let GXO. 
, &c. represent the coefficients of the wth power of (1 -f w)., 
so that 
^ /n\ n.n — 1 
Vo;=^’ Or 
2 / 1.2 
n.n — l.n — 2 
1.2.3 
&c 
then 
( n — 1 / n 
-““1:2” ’ - T n 
