282 BELL SYSTEM TECHNICAL JOURNAL 



So by taking 



r = 5 (mod 2«i), 

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

 k = 2, 3, A, •■•,t 

 it is concluded immediately that 



^x(JV) = 1 (modiV), 



r^ ^ dz 1 (modiV), b < \(N). 



The preceding formal analysis for Case I and Case II may be 

 summed up as having established the following general theorem : 



If N is a given positive integer and if s is an integer prime to N, then the 

 largest possible exponent dfor which the companion congruencial relations 



s'^ = ±\ (modiV), 



s^ ^ ±\ (mod N), b < d 



will be true is X(iV)/2 for numbers such that X(7V) = <t>{N) and is \{N) 

 for numbers such that X(iV) < (l){N), and a value for s belonging to this 

 exponent in each instance does exist. 



In order to apply the foregoing results to a practical case Table I 

 has been prepared. In the left-hand column appear the numbers 5 to 

 139, inclusive. In the next column is listed for each number the value 

 of X(iV)/2 or of X(iV), depending upon whether \{N) = (p(N) or \{N) 

 < <i>(N). In the final column there is listed for each number a suitable 

 value for the spread. There appears to be no advantage of one spread 

 figure over another, and the listing of additional acceptable values is 

 omitted in the interest of economy of, space. For the numbers for 

 which X(iV) = 0(iV) and for which \(N)I2 is odd care has been taken 

 that the listed spread figures are primitive roots, and not the squares of 

 primitive roots which were shown to be equally acceptable. This fact 

 will be recalled later. 



It was shown earlier that \^{N — l)/2] successive cable lengths 

 would be the maximum possible number for an extended conductor 

 unit to traverse without incurring repetition of at least one of the same- 

 layer adjacencies which occurred in the first of these lengths. On 

 referring to Table I it is seen that only for the prime numbers is this 

 maximum attainable. The prime numbers are distinguished by the 

 fact that for them X(JV)/2 = (N — l)/2, and each has been indicated 

 by an asterisk. The composite numbers are seen to yield quite in- 

 ferior results in general. 



