314 BELL SYSTEM TECHNICAL JOURNAL 



Congruences 



For numerical checks, it is convenient to note the simplest con- 

 gruences ^ for the three numbers. These follow from the Touchard 

 congruence for the e numbers ^^ which runs as follows: 



ep+„ = e„^.i + €„ mod p, 



where /> is a rational prime greater than 2. 



Since by equations (10) each of the impedance numbers is a linear 

 function of the e numbers, each has a similar congruence as follows: 



Dp+„ = D„+i + Dn mod p, 



Tp+n = Tn+i + Tn mod p, (11) 



Up+n = Un+\ + Un mod p. 



Special values for the first few congruences are as follows: 



Remainder, mod p 



These are sufficient for checking every value in Table I at least 

 once and the values for w = 5, 6, 7, 8 are checked twice. 



Acknowledgment 



This paper arose as a result of a suggestion made by R. M. Foster 

 on a former paper '^ and thanks are also due him for continuous 

 counsel and critical scrutiny which have enlarged the boundary and 

 sharpened the outline of the problem. 



* The congruence Dn = r mod p is equivalent to the equation Dn = mp + '', 

 where m is an integer; that is, r is the remainder after division by p (or the remainder 

 plus some multiple of p). 



"See E. T. Bell, "Iterated Exponential Integers, "Annals of Math., 39, 3 (July, 

 1938), eq. 1.101, p. 541. 



11 "A Ladder Network Theorem," Bell System Technical Journal 16, pp. 303-318 

 (July, 1937); see especially footnote 3; I take this opportunity to draw attention to 

 an error in that footnote: for four terminals (see Table I) there are 157, not 64, 

 measurable impedances; hence the upper bound to the number of representations is 

 18,883,356,492, not 74,974,368. 



