CLASSIFICATION AND DETERMINATION, etc. 23 



which requires r = 0. Having v = qm, we inquire under what con- 

 ditions does q = A? Since 



^x pnv 1 __ d pnmg1 



ei " T ' ~^r=ir' 



it follows that A divides (p nm< i- l)/(p_ 1). Raising to the power q 

 the identity p nm = 1 + (p nw - 1), we find 



Let be a prime factor of A and a the highest power of con- 

 tained in A. Since 6 divides p nm 1 and the left member of 12), it 

 must divide q. Further, if > 2, Q a divides q. Indeed, the ratio of 

 the & tb term of 12) to the first term q can be written 



of which the first two factors are integers, while the third factor 

 {i -He- !)} 



- _ 



k 



is > 1 if k ^ 2. Hence the irreducible fraction equal to 6 k ~ l /1c has 

 the factor in its numerator. Hence the terms of 12) beginning 

 with the second contain to a higher power than the first term q. 

 Since a divides A, which divides the left member of 12), it follows 

 that a divides the first term q on the right. Hence, if A be odd 

 or the double of an odd number, q is divisible by A. Inversely, if q 

 be divisible by A, A being odd or the double of an odd number, the 

 above argument shows that the right member of 12) will contain 

 the factor A and therefore that the left member of 11) will be an 

 integer. In order that v be the least integer for which this can 

 happen, we must have q = A. 



If A be a multiple of 4, p nni 1 is even by hypothesis. Then 

 = 2 will be a factor of q as before. The ratio of the second term 

 of 12) to the first term will be divisible by 2 if, and only if, p nm 1 

 be a multiple of 4; the ratio of the & th term to the first will, for k ^ 3, 

 contain the factor 2. Hence, if p nm be of the form 41 + 1, we can 

 conclude that q = A. [The case p nm = 4Z 1 leads to the entirely 

 different theorem of 36.] 



35. Let ^ be a primitive root in the GF[p n ]. The function x Q* 

 belongs to the exponent (p n l)/d where d is the greatest common 

 divisor of t and p n 1. Applying the theorem 34 for m = 1, we 

 have the result: 



If A ~be any integer containing no prime factor not occurring 

 in p n \ and if t ~be an integer prime to A, the 7[A, p n ~\ belonging 



