FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX, 
203 
If the second exponent be a 2, the third may be also, and so on ; the series then 
continues with 2^, 2®, 2^, . . . Of the 2’s there are at least two ; for otherwise the first 
three exponents would be = 1, = 2, fg = 4, making 4 an exponent for mod 2® = 8, 
which is impossible. 
Hence the series of exponents run either thus— 
or thus 
1 . 1 . 1 . . . . 1 . 2 . 2 ^ 2 ^. 2 ^ . . . 
1. 2. 2. . . . 2. 2. 2n 2l 2‘* . . . 
These results can be expressed thus— 
Let the highest power of 2 which divides a- — 1 be (Since a is an odd 
number, s is at least equal to 2.) 
Then the exponent of a is 
if \>s, 2^-* 
if 
2 if cr = 1 (mod 2^) and « ^ 1 (mod 2^). 
1 if a = 1 (mod 2^). 
Corollary. —The greatest exponent possible for mod 2^ is 2^“"; and as </>(2^) = 2^"b 
primitive roots do not exist. 
Examples .—Exponent of 3 for mod 2'’. 
and therefore 
Exp of 3 for mod 2 is 1, 
„ ,, 2^ is 2, 
2Hs 2, 
„ „ 2Ms 4, 
exp of 3 for mod is 145 
Exponent of 35 for mod 
therefore 
therefore 
(54 = 2^' (X = G). 
352 — 1 = 1224 = 153.23, 
5=2, 
exp of 35 is 2'*'. 
Exp of 41 for mod 128 = 2'^ (X = 7). 
4 H - 1 = 1680 = 105.2h 
2 D 2 
