FOR ARY COMPOSITE MODULUS, REAL OR COMPLEX. 275 
Example .—Mod (1 + ‘I>(l + ^y = 2®. The three generators have exponents 
2 ^ 2 ^ 2 ®. 
The 2'^ + 2® numbers with exp 2® are (see Proposition xv.)— 
^7~j“2'i, 5“}"2?, 6-hL S?-, 3-T'22., l-j-2i, 2-j~32., 2-|~i[|mod (1 
The 2® + 2“^ + 2® numbers with exp 2^ are 
= dz ^ [mod (1 + and 7, 9, 3 + 4i, 5 + 42, 11 + 42, 13 -f- 42 [mod (1 -j- 2 ')^]. 
The 2^ + 2 + 1 numbers with exp 2, are 
15, 17, 31, 7 + 8 ?, 23 + 82 , 25 + 82 , 9 + 8 ?. 
If of these we take 
2 + 2 with exp 8 , 
3 with exp 8 , 
and 
we have 
(2 + 2)^ 
34 
9 + 82 
;17 
15 
2 with exp 4, 
> and 15. 17 . (9 -{- 82 ) 7 82 ^ 1 . 
Therefore 2 + 2, 3, and 2 generate the 2® numbers. 
(xvii.) From Propositions ix., xii., xiii., and xiv. the exponent of a number for any 
modulus (to which it is prime) is readily determined. 
Let the modulus be expressed as a product of powers of its prime factors, 
m = (1 + 2)'‘P/‘P/^. . . 
where P;^, Pg, . . . are pure primes, and Q], Qg, . . . are mixed primes, and equal to 
“1 + /3iL ag + ^g2 . . . 
Then, by Proposition ix., the exponent of any number a is the L.C.M. of its separate 
exponents for the moduli 
(1 + iy, Pl^ Pg^ . . . Q/s Qg'^^. . . 
The greatest exponent -possible .—The greatest exponent for mod (1 + 2)" 
is 
1 if K = 1 , 
2 if K = 2 , 
4 if 2 < /c < 8 , 
2*' - Mf 8 ^ /c = 2k or 2k + 1. 
2 N 2 
