FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
201 
If, then, the exponent of a for moclulas p he ti= t and a* — 1 contain as the 
highest power of p, we have 
cd = 1 (mod p^), 
and 
^1 = ^3 = • • • = fs= h 
And after these 
+ 1 - P^S) 
^s + 2 — + 1 ) 
&c. 
Hence the exponent of a for modulus p)^ is 
t if X < s, 
tp^~^ if X > s, 
where t is the exponent of a for modulus p, and p^ is the greatest power of p that will 
divide cd — 1. 
Corollary. —The greatest value that t can have is — 1. This is so when a is 
congruent to a primitive root of p) (Proposition 11, corollary). The greatest value 
that^/"® can have is got by making s as small as possible, viz., by making 5 1, i.e., 
by taking a so that — 1 (though necessarily divisible by y>) is not divisible by p^^. 
Therefore the greatest possible exponent that a number can have for modulus pp^ is 
{p — 1) • p^~^, and as this is equal to {p’^) it follows that primitive roots exist for a 
modulus a power of a prime. 
Examples. —Exponent of 3 for mod S''. 
Therefore 
The exp of 3 for mod 5 is 4 (3. 4. 2. l). 
3^ — 1 = 80 is divisible by 5h 
exp of 3 is 4. 5^ (mod 5®). 
Exponent of 24 for mod S'’. 
Therefore 
The exp of 24 for mod S is 2 (^ = 2). 
24^ — 1 = 57S which is divisible by S’b {s 
exp of 24 mod S® = 2. Sh 
= 2 ). 
(13.) The exponent to which a number a belongs for a power of 2 as modulus. 
The number a is now to be considered odd. 
Let ij, t. 2 , ... ... he the exponents of a for moduli 2, 2~, . . . 2'^ . . . 
MDCCCXCIII.—A. 2 I) 
