FOR ANY COMPOSITE MODULES, REAL OR COiMPLEX. 
191 
Hence the term Avhich hi st appears a second time is the first term a ; whicli 
appears next as t being the least number for which 
rfi = 1 (mod m). 
Defi'iutiun .—The smallest number t which makes oJ = 1 (mud lu), where a is piime 
to m, is called the exponent of a for modulus ni. (Jau('1IY uses the word “ iiaficalon'' 
in the same sense. He also uses “ niaximuni indicaton ” to denote what will be called 
the highest exponent. 
Thus the infinite series of residues of a, cd, id, . . . consists of a repetition of the 
period of t terms beginning from the first term. 
(2.) If t be the exponent of a and cP = 1 (mod ni) then t divides s. 
Let 
s = ip + where r < t. 
Then 
<_e •— ^ I (mod ni), 
[np . (X = I (mod ni), 
iC = 1 (mod io), 
whereas t is the smallest value (not zero) which makes u} — 1. I'herefore 
therefore 
(3.) Fermat’s Theoreni. 
/•= 0 , 
t divides s. 
f-pim) ^ I (mod lit). 
Let rtj, Oj, ttg, . . . be the (j) (m) numbers less than in and prime to it. 
Take any one of them a. 
Then since aa.^, aa.^, . . . aa^(/„)(mod m) are all prime to ni, and no two con- 
gruent, they must be the same set of numbers as Oo, . . . therefore 
and, therefore, 
apip.^ . . = aph • • • (gnod in), 
^ I (mod m). 
Corollarij .—It folloAvs from this proposition and proj)osition (1) that the exponent 
ol any number (modulus in) is always a divisor of (ji^ni). The pro])ositions which 
follow will determine that divisor. 
(4.) If t be the exponent of a then the exponent of (P is r : where t = kt and k is 
the G.C.M. of s and t. 
