FOR ANT COMPOSITE MODULUS, REAL OR COMPLEX. 
229 
and thus 
a = {fio + (To^o + -f • • •)'“ (^0 + 71^1 + . • • • • (mocl m), 
and all the factors in this product (omitting those whose y is = 1) form a complete 
set of 2-power-exponent generators. 
Thus, in either case, when p is an odd prime or is equal to 2 we can form a set of 
p-power-exponent generators (having the exponents found to be necessary in 
Proposition 26), such that each is congruent to unity for all but one of the principal 
factors of m. 
Example .— Let 
m = 308 = 23. 7. 11. 
<>, (m) = (2). (2. 3) (2. 5), 
The highest exponent is 30. 
There are 
7 numbers with exp 2 '| 
and > (Example Proposition 26.) 
1 >, .. 1 J 
We will form unitary generators of these eight numbers. Since 2 enters only in 
the square into m, fm not needed. 
We must take 
Jq— 3 (with exp 2 mod 23), 
yi= 6 (with exp 2 mod 7), 
y.,= 10 (with exp 2 mod 11). 
Thus 
^ 0 = 77 , ^ 1 = 176 , ^ 3 = 56 . 
S'o = ( 3 - 1) 
cj^ = { 6-1) 
^2 =( 10 - 1 ) 
77 + 1 = 155 (mod 308) 
176 -f 1 = 881 = 265 (mod 308) 
56 -f 1 = 505 = 197 (mod 308) 
> 
are a (in this case the) set of 2-power-exponent unitary generators. 
The numbers with exponent 2 are given as products of powers of these by the 
following table of indices. 
Numbers, 
Index of 155 
Index of 265 
Index of 197 
197, 265. 155. 111. 43. 
0 0 1 11 
0 10 10 
1 0 0 0 1 
153. 307 
0 1 
1 1 
1 1 
