NUMBER THEORY IN SPLICING OF CABLES 277 



values for N less than 5 are of no significance to this problem. In 

 the analysis which follows, therefore, no particular effort has been 

 made to render the general conclusions capable of extension to these 

 extreme and trivial cases. 



It is necessary at this point to recall and introduce certain working 

 material. First, there is the established theorem that every positive 

 integer N greater than unity can be represented in one and only one 

 way in the form 



N = pf'pi"^ • • • pt"t, 



where pi, p2, • • • , pt are different primes and ai, 0:2, • • - , at are positive 

 integers. Then there is the familiar number theory function (t>{N) 

 which indicates the number of positive integers not greater than N and 

 prime to N* If ^ is a prime number and a: is a positive integer, then 



<i>(P") = P^-KP - 1); 



also 



<i>{pf'pf' ■ ■ ■ pt^O = 4>{pf')-4>{pi'''-) </>(^t"0, 



where pi, p^, • • ' , pt are different primes. 



Then there is the X-function defined in terms of the (^-function as 

 follows: 



X(2«) = 0(2") for a = 0, 1, 2, 



X(2-) =^^fora > 2, 



X(^") = 0(p") for p an odd prime, 

 \{2"^p>fipf^ . . . p^at) = M, 

 where M is the least common multiple of 



x(2«o, x(^2"^), KPz"'), ■■', HpfO, 



2, p2, p3, • • ■ , pt being different primes. f Finally, it is established that 

 for two relatively prime integers 5 and N the value X(iV) is the largest 

 possible for the exponent m for which the relations 



^'^ = 1 (mod N), 



s" f^ 1 (mod N), n < m, 



* Euler, "Novi Comm. Ac. Petrop.," 1760-61, p. 74. Carmichael, "The Theory 

 of Numbers," John Wiley & Sons, Inc., 1914, pp. 30-32. Dickson, "Introduction 

 to the Theory of Numbers," Univ. of Chicago Press, 1929, Chap. I. 



t Cauchy, Comptes Rendus, Paris, 1841, pp. 824-845. Carmichael, p. 53. 



