NUMBER THEORY IN SPLICING OF CABLES 279 



value for d greater than \{N)I2 is possible is evident immediately. 

 For let a be any integer prime to N. Then for some exponent k 



r'' = a (mod N) and 



aXW/2 = ykHN)l2 = _[_ 1 (mod A^). 



Next there will be considered the companion relations 



(7) 5'' = 1 (mod TV), 



(8) 5'' ^ ± 1 (modiV), b <d. 



The reasoning just above shows that d cannot be greater than X(iV)/2. 

 Suppose for the moment that d has this greatest possible value \(N)/2. 

 Relations (7), (8) then become 



^x(Ar)/2 ^ 1 (modiV), 



5^ 7^ ± 1 (modA^), b = 1,2, 3, ■••,MN)I2 - I. 



These relations may be written 



(9) (5i/2)xw = 1 (mod N), 



(10) (51/2)25 ^ ± 1 (modiV), 2^ = 2,4, 6, •••,X(iV) - 2. 



Now relations (9), (10), will be compatible with relations (1), (2), (3), 

 (4) only if \{N)/2 is an odd number, for otherwise the restrictions of 

 relation (10) applying to the even numbered exponents from 2 to 

 X(iV) — 2 inclusive would be in conflict with relation (3). For X(iV)/2 

 an odd number, then, relations (9), (10), are satisfield for s^i^ a primi- 

 tive root of N. Consequently, with relations (7), (8), X(iV)/2 is the 

 largest possible value for the exponent d, and a value for s equal to the 

 square of a primitive root of N permits this to be attained. 



Case II. \{N) < (^(iV). 



The inquiry for this case will be divided into four parts. In general 

 N = pi"^p2"-pi"^ • • ' pt"^ where pi, p2, pz, • - • , pi are difi'erent primes. 



(a) First will be considered the case where pi, p2, ps, • • • , pt are all 

 odd primes. Then X(A'') is the least common multiple of \{Pi°^), 

 \{p2"'^), \{pz"^), • • • , \{pt"^). Suppose now that the highest power of 2 

 dividing any of the X's divides X(/)i"'). If this same power of 2 divides 

 more than one of the X's, arbitrarily select \{pi"') as one of them. Then 

 this power of 2 will be exactly that occurring in X(A^). Now arbitrarily 

 select pj as any one of the odd primes other than pi. Then clearly 



