Rmjal Society, 71 



theorem, and to give the cases in which 1 is to be added or sub- 

 tracted ; and in the course of the proof, he mentions that the num- 

 bers prime to any number not only are found in pairs, one greater 

 and one less than one-half of the number, but that they associate 

 themselves in sets of four, with an odd pair in certain cases. Thus, 

 the primes to 7 are 1, 2, 3, 4, 5, 6, — 



2x4=8=7 + 1. 



Put the complemental numbers underneath crosswise, thus, — 



2x4 



\/ 

 / \ 



y \ 



3X5 

 80 that 2 + 5 and 4+3 may equal 7 ; and then 



3x5=15=2x7 + 1 



2x3= 6=7-1 4x5=20=3x7-1 



Multiplied together one way the product exceeds 7, or a multiple 

 of it, by 1 ; multiplied the other way, the product is less than 7, or 

 some multiple of it, by 1. By assuming the prime number to be A, 

 and the two primes to it to be jo, q, and that p-\-q he not equal to 

 A, but j05'=wA+l, it is shown that the complemental primes 

 {A—q) and (A—/?) will have a product=w'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=7w— 4' ; 



therefore ] .2.3.4.5.6+43=7^; 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 l,p, q, r, &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 z^+ 1 must be divisible by A, and there- 

 fore that z"*- 1 must be divisible by A. If A be a prime number 



