666 



(A— §') and {A—p) will have a product=?^'A + l, and that, in- 

 stead of 1, the number may be any other prime to A. Upon this 

 foundation the author proceeds to show that Wilson's theorem, and 

 also Gauss's, may be made much more general ; that if A be a prime 

 number, as 7, the numbers less than it may be arranged in pairs, 

 not only with reference to 1, but to any number less than 7. Take 

 4 as an example : — 



1 X 3=7-4 



4 X 6=4x7—4 



2 X 5=2x7-4 

 therefore 1 .2.3.4.5.6=7^— 43; 



therefore ] . 2 . 3 . 4 . 5 . 6 + 4^ = 7w ; that is, is divisible by 7. 



The same is then shown as to numbers not prime, provided those 

 numbers alone are taken which are prime to it, and the number of 

 pairs will be half the number of primes. The general theorem 

 therefore is this :— If A be any number, prime or not, and m be the 

 number of primes to it, which are &c. ; then 1 .p .q.r, &c., 



m 



+ Z2 will be divisible by A, provided Z be prime to A, whether it 

 be greater or less. 



It follows from this that s^f + 1 must be divisible by A, and there- 

 fore that 2;"*— 1 must be divisible by A. If A be a prime number 

 and z a number prime to it (which every number not divisible by it 

 is), this is Fermat's theorem, and the author has given a new proof 

 of it. But the theorem is true though A be not a prime number, 

 provided z be prime to A and m be the number of primes to A, 

 and less than it; and instead of 1, any other number prime to A 

 raised to the mth power may be substituted : and 2;"'— 2/"' will be di- 

 visible by A, provided z and y be primes to A, and m be the number 

 of primes to A and less than it. 



The author has therefore in this paper offered a proof of Gauss's 

 theorem, and proved that it applies in certain cases to one half of the 

 primes, and in all cases, with certain modifications, has shown that 

 a similar property belongs to the product of the odd numbers, and 

 also of the even numbers which precede any prime number; and 

 lastly, has shown the intimate connexion between Wilson's theorem 

 and Fermat's, and shown that each is but a part of a much more 

 general proposition, which, he observes, may itself turn out to be 

 part only of a still more universal one. 



In a postscript, the author has shown that the well-known law of 

 reciprocity of prime numbers is an immediate corollary from his 

 theorem ; and that it may be extended thus : if A and B be any 

 two numbers (not prime numbers but) prime to each other, and the 

 primes to A, and less than it, are \m) in number, and the similar 

 primes to B are (n), then (A*"— 1) is divisible by B, and (B'"— 1) is 

 divisible by A. 



