280 BELL SYSTEM TECHNICAL JOURNAL 



X(iV) will also be the least common multiple of 



Now take 



r = gj^ (mod pi"')> Si ^ primitive root of pj"', 

 = gk (mod pk"''), gk a primitive root of pk"'', 



k = 1,2,3, •••, j - l,i+ 1, •••,/. 



The r thus chosen must be prime to each of the prime factors of N, 

 and hence must be prime to N. Consequently it is known that 



rMN) = i (modiV). 



Suppose that m is the smallest exponent for which the congruence 



r*" = 1 (mod N) 



is true. Then it is noted that the chosen r is such that m must be a 

 multiple of 



XC/'i-O, X(/'2-), •••, X(^,-_i«'-0, MM!), x(/,,.^,«y+i), ..., X(^,«0. 



and the least multiple common to these is, of course, X(iV). Therefore 

 it can be written that 



rMN) = 1 (jnod N), 



r^ ^ I (modiV), b<\(N). 



Now suppose that for some exponent n less than XiN) 



r" = - 1 (mod N). 

 Then 



r^n = 1 (modiV), 



and if n is less than \{N), 2n is less than 2\(N) and can only be equal to 

 X(iV). It would necessarily follow then that 



and it would follow in turn that 



rMN)i2 = _ 1 (mod />;«'■)• 



