278 • BELL SYSTEM TECHNICAL JOURNAL 



will hold, and that a value for 5 belonging to this exponent does exist.* 

 Here it is convenient to consider separately numbers of the two 

 classes — those for which X(iV) = <t>{N) and those for which \(N) 

 < <f>{N). For numbers of the first class established theorems may be 

 drawn upon to furnish a complete analysis. For numbers of the 

 second class, however, it will be necessary to extend a bit beyond the 

 ground covered by previous workers, and the steps will be given in 

 considerable detail. This procedure coupled with the inherent com- 

 plexity will render the treatment for the latter class much less compact 

 and elegant than that for the former. 



Case I. \{N) = <I>{N). 



From the defined relation between the ^-function and the X-function 

 it follows that numbers of the class such that X(iV) = <^(iV) are con- 

 fined to the values 



1, 2, 4, p", and 2p", 



where p is an odd prime and a is a positive integer.f For a number N 

 of this class it is established that there exists a set of (^((^(iV)) numbers 

 r, such that 



(1) r^(^) = 1 (mod N) and 



(2) r" ^ 1 (mod N), n < \{N), \(N) = 4>{N). 



Such a number is known as a "primitive root" of N.X From the 

 properties of the primitive root r of the number N as defined by rela- 

 tions (1), (2) it follows readily that 



(3) r^(^^)/2 = - 1 (modiV), 



(4) r" ^ ± 1 (mod N), <n < ^^ , ^^ <n < \{N). 



First there will be considered the companion relations 



(5) s'^ = - \ (mod A^), 



(6) 5* ^ ± I (modiV), b < d, 



and, from comparison with relations (3) and (4), these clearly are 

 satisfied for s a primitive root of N and for d = \{N)/2. That no 



* Carniichael, p. 54 and pp. 61-63. 

 t Carmichael, p. 71. 



j Gauss, " Disquisitiones Arithnieticae," Art. 52-55. Carniichael, pp. 65-71. 

 Dickson, Sec. 17. 



