MR H. r. TALBOT ON THE THEORY OF NUMBERS. 47 



And therefore all the terms of the congruence may be omitted except the two first. 



.•.«^~VZ^O, whence -«^-^ ^ Z. 



In other words, if Z is divided by p, it leaves the same remainder as «^~^ does, 

 when divided by p, but with a contrary sign ; x being any one of the j9- 1 numbers, 

 which are less than p. 



The simplest case is when x=l. In this case the theorem gives — 



-l^Z orZ + 1^0(mod.p); 



which result, expressed in other words, is : — " If p be any prime number, the 

 product of all the numbers less than p, or 1, 2, 3, ... . {p—V), augmented by 

 unity is divisible by j9." 



This is the celebrated theorem, known as " Wilson's Theorem," of which 

 neither its inventor nor Waring, who first published it, could find any demon- 

 stration. It was first demonstrated by Lagrange (Berlin Memoirs, 1771). 



We have not employed Fermat's theorem in demonstrating it, therefore it is 

 well to show that the latter can be deduced from it. Thus, we have found 

 x^-'^^ — Z (mod.j9). 



But we have found Z ^ — 1. And therefore x^~'^ ^\ {x being any number 

 less than p)^ which is Fermat's theorem. 



§ 2. On Associate Numbers. 



By Wilson's theorem, the product of all the numbers 1, 2, 3, ... . (^9-1), is 

 congruent to -1 (mod. p). Another demonstration of this is given in Gauss's 

 " Arithmetical Researches" (French translation, p. 57). It is there said that 

 EuLER discovered that this product, omitting the first and last numbers 1 and 

 p— 1, could be divided into pairs of associate numbers, the product of each of 

 which is ^ 1 (mod.^), while the product of the remaining two numbers, 1 and 

 p—\ is obviously ^ — 1 (mod. p). So that the product of the whole series 1, 2, 

 3, . . . . p—l, is ^ — 1 (mod. p))^ as we found before. 



In the passage quoted, the following example is given: — The numbers less 

 than 13 can be multiplied in pairs, thus : — 3 x 9=27 = 1 (if we omit the multiples 

 of 13), which we write 3x9^1 (mod. 13). 



Also, 2 k 7 EE 1, 4 X 10 EEE 1, 5 X 8 :ee 1, and 6x11 = 1. But, on the other 

 hand, 1 x 12 ^ — 1. Therefore the whole product 1, 2, 3, .... 12 ^ — 1. 



In this theorem of Euler's, the product of each pair ^ 1, with the excep- 

 tion of one pair, which is ^ — 1. 



I have found that there exists another and very different system of associate 

 numbers, in which the product of each pair is ^ — 1 ; and therefore, the product 

 of the whole is ^ — 1 whenever the number of pairs is odd ; but if it is even, 

 in that case the product of one pair always deviates from the rule governing 

 the rest, and is ^ + 1. So that in all cases the product of the whole is ^ — 1. 



