2 REPORT — 1856. 



Taking, for example, the modulus 7. the congruence 

 na; = w+l (mod. ^) 

 is a type of the six congruences (mod. 7), 



1 .2 = 2 



2.5 = 3 



3.6 = 4 

 4 .3 = 5 

 5.4 = 6 



in which, while to n are given successively the values 1 . 2 . 3 . 4 . 5 . 6, we give to x 

 the corresponding values 2, 5, 6, 3, 4, 7. 



From this simple theorem Mr. J. T. Graves derives Wilson's famous theorem, 

 namely, — 



" When ^ is a prime number, we have 



1.2.3 (p— 1) = -1 (mod.p)." 



It is easy to see that the congruence (;)— l)a;=^ is solved by making ce:=p, and 

 hence, by the preceding theorem, it is possible to find among the quantities 2.3.4.., 

 p— 1, distinct values, including all numbers from 2 top — 1, for x^, x^, x^, . . Xp-z, 

 such that 



1 .ari = 2 



2.a;3 = 3 (a) 



3.«3 = 4 (b) 



{p—2)xp-2z^p—l. 

 If, as is allowable, we substitute 1 . x-^ for the factor 2 in the left-hand member of 

 congruence (a), we get 



l.iri.a;j=3 (c) 



Again, if we substitute 1 . a^i . a?j for the factor 3 in the left-hand member of con- 

 gruence (b), we get 



and proceeding similarly, we find 



1 .a?i . Xj . iTg . . . arp-2 = /'— 1 = — 1. (A) 



but by Mr. J. T. Graves's theorem, 



1 .Xi.Xi.Xi a-p-2=1.2 .3 (p— 2)(jp~l). 



Hence we have by (d), 



1.2.3 {p—'i){p-\) = — l. Q.E.D. 



For example, with respect to modulus 7, we obtain in this manner the six con- 

 gruences, 



1 = 1 



1 .2^2 



1 .2 .5 = 3 



1 . 2.5.6 = 4 



1.2.5.6. 3 = 5 



1.2 .5 .6.3.4 = 6, 



the last congruence being equivalent to 



1.2.3.4.5.6 = — 1 (mod. 7). 

 Wilson's theorem is thus exhibited as the last of a series of minor theorems. 



