FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
205 
or by 
Exp 1. 2. 2. . . . 2. 22. ... 2 ^^ 
Mod 2. 22. 23. . . . 2^-^+b 2'^-’ . . . 2"J 
In either case 
cr = 1 (mod 2''~^'^ ^) 
cr ^ 1 (mod. 2''“ ® 
therefore 
a® = 1 + + ^ (mod 
and therefore 
« = ± 1 + 2* “ * (mod 2’'" ® *). 
Thus the numbers (modulus 2*) which have exponent 2* are the 2* numbers 
given by 
d: 1 +2''“® (mod + 
In particular, the numbers with the greatest exponent 2''“^ are the 2''“" numbers 
±1 + 2^ (mod 2^). 
We have seen that there is one number (viz., unity) with exponent 1, 
there are 3 numbers with exponent 2. 
93 
?? 
? J 
5 5 
22. 
9K -2 
?5 
55 
55 
9/C - 2 
In all, 1 + 3 + 2^ + 2^ + . . . + 2*”“ = 2'‘~* = <^(2''), the number of odd numbers 
less than 2*, as it should be. 
(16.) If we take any number (j which has exponent 2''"“ for modulus 2*, the 
successive powers of g give 2''~ - incongruent numbers, one half of the complete set of 
odd numbers less than 2''. 
Of these, one and only one, has exponent 2, viz., g~''~^ (Prop. 4). 
Now 
squaring. 
and successively squaring 
1 + 2 ^ 
ry"" = 1+2^ 
(mod 23) 
(mod 2“*) 
(mod 2®) 
gV ^ _ 1 + 2'' ^ (mod 2*), 
