NUMBER THEORY IN SPLICING OF CABLES 281 



However, r has been chosen such that 



yXW/2 = (g.2)X(N)/2 = gUN) = 1 (j^oJ p^ai^ 



This last relation is incompatible with the one immediately above, and 

 it must be concluded that the assumption 



r" = - 1 (mod N), n < \(N) 



is false, and that for the r that has been chosen 



rMN) = I (mod N), 



r& 7^ ± 1 (modiV), b < \{N) 



and no exponent greater than \{N) is possible. 



(b) Next will be considered the case where pi = 2, ai = 1, and 

 p2, p3, ' ' ' f pt are all odd primes. Select pi as above and take pj 

 different from 2. Then take 



r = 1 (mod 2), 

 = gj^ (mod pi">)r gk a primitive root of pf', 

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



^ = 2, 3, 4, --^i - l,i+ 1, •••,', 



and the same line of reasoning may be repeated and the same con- 

 clusions reached as under part (a) above. 



(c) Next will be considered the case where ^i = 2, ai = 2 and p2, 

 Pz, ' • • ,pt are all odd primes. Since X(22) = 2 take pi different from 2, 

 and for simplicity take pj as 2. Then take 



r = 1 (mod 4) , 

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

 k = 2, 3,4, ■■■, i 



and the same line of reasoning may be repeated and the same con- 

 clusions reached as under part (a) above. 



(d) Finally will be considered the case where pi = 2, ai > 2, and 

 p2, pz, ' • • , pt are all odd primes. Now 5 has the property that 



5X(2«i) ^ 1 (mod 2«0, 



5* ^ ± 1 (mod 2°i), «i > 2, b < X(2«0- 



