sy — ae 
a . is . 
® < 
Interesting Properties of Numba ~ i188 
mainders must recur in periods whose number of term cannot 
exceed P-—1; for there can be but P—1 different remain TS5 
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 Fermat, that N?-' divided by P will 
leave 1 for remainder, that is7p_,=1. Hence we conclude, that 
T= TesP-, (7), also Ye=Fx4P-} (8). 
It also follows, that when the number of terms in the periods 
of quotients and remainders is less than P —1, it must be a sub- 
multiple of P—1. 
Suppose we should find Tp__,=P—1, then the remainder 
n 
‘p_; _ Will be found by dividing NP—N by P, or simply by 
——+1 
. n 
dividing —N by P ; we have already indicated the remainder of 
N divided by P, by 7, ; therefore the remainder of — N by P will 
—? 1, or more correctly P—r,. Hence r, _ bdeyin xi or 
"p_y _+7,=P; after the same manner we prover, _, + 
——+1 = 
n 
r.=P (9), 7 
From the general equation of (1) we get Pg, =Nr._, —r., (10). 
Changing x into P— tie we have Pgp_y=Nr 
wai 
= P-1, 4 
n n 
~"p_y (11). Taking the sum of (£0) and (11) and reducing 
=a} S, 
it, 
by means of (9), we get Yp_1. +9. =N-1, (12). There- 
4 paint: 5 
n 
fore, whenever the remainder rp_y=P-1 the number of terms 
n 
: : e : 24P—1 
'n the periods of quotients and remainders will be — and 
Vol."xt., No. 1.—Oet.-Dec. 1840, 15 
