FOR ANT COMPOSITE MODULUS, REAL OR COMPLEX. 
283 
Hence the number of numbers with exponent modulus m is 
Example .—Mod 3^(3 + 2i). The greatest exponent is the L.C.M. of 3^(3’^ ~ 1) 
and 12, = 72. 
We will find how many numbers have exponent 3^, and how many numbers have 
exponent 3. 
fi>(33) = 23.3^ 3x 
(I>(3 + 2f) = 2^ 3. 
The number of numbers with exponent 3^ is 
g(3 + 2' + l)2 __ g(2+2 + l)i 
= 35 - 33. 
The number of numbers with exponent 3 is 
Hence 
and 
0(2 + 3 + l)i Y 
33 - 1 . 
33 — 33 numbers have exp 3^. 
33 — 1 numbers have exp 3. 
1 number has exp 1. 
(xxiiirt.) By writing $(1+7)''in the form 2''“2''“'2''"" we are enabled to apply 
exactly the same method to this case as we have to the case of any odd prime The 
result we obtain is that the number of numbers with exponent 2'* for mod m is 
2(2/<)s __ 
Example .—Mod 12 + 47 = 7 (1 + 7)3(1 + 27). Highest exponent = 2^. $ (m) = 2®. 
$(1 + 7)3 = 23 . 2 . 2 . 
$ (1 + 27) = 23. 
The number of numbers with exponent 2^ 
+ 1 + 1 + 2)2 “ (2 + 1 -f* 1 + 2)i 
— 2^ — 2b 
2 o 2 
