Interesting Pi'&perties of Numbers. 113 



mainders must recur in periods whose number of terms cannot 

 exceed P - 1 ; for there can be but P — 1 different remainders ; 

 so that if we extend the process beyond P— 1 terms, we shall be 

 sure to fall upon a remainder like one that has already occurred, 

 and then the quotients and remainders will begin to repeat. 



Thus far our conclusions have been general, that is, they are 

 correct for all values of N and P. We will now deduce some 

 properties which hold for particular values of N and P. 



When P is a prime, and N is not divisible by P, we know by 

 the celebrated theorem of Perfuat, that N^"" ' divided by P will 

 leave 1 for remainder, that is rp_j =1. Hence we conclude, that 

 r, = r,+P_, (7), also g. = g',+P_, (8). . 



It also follows, that when the number of ter'ms in the periods 

 of quotients and remainders is less than P — 1, it must be a sub- 

 multiple of V—1. 



Suppose we should find ^p_j=P — Ij then the remainder 



n 

 rp_ I will be found by dividing NP— N by P, or simply by 



n 

 dividing — N by P ; we have already indicated the remainder of 

 N divided by P, by r, ; therefore the remainder of - N by P will 

 be —r,, or more correctly P — r,. Hence r^ ^ =P— r, or 



^— i + 1 



n 



Tr) 1 -f Tj =P ; after the same manner we prove ^p _ ^ + 

 f-Zi-fl ^—l+x 



n n 



r.=P (9). 



From the general equation of (1) we get P^'^^Nr^.i — r^,(10). 



Changing x into -{-x we have P^-p _ -. =Nrp_ ^ 



'>^ -\-x -{-x—1 



n n 



— Tp ^ (11). Taking the sum of (10) and (11) and reducing 



by means of (9), we get 9'p_ j -Vq. = N - 1, (12). There- 



-\-x 



n 



fore, tvhenever the remainder ?'p_ i =P — Ij ^he number of terms 



n 



2(P — 1) 

 in the periods of quotients and remainders icill be -^ 1 and 



Vol.>L, No. 1.— Oct.-Dec. 1840. 15 



