FOR A^Y COMPOSITE MODULUS, REAL OR COMPLEX. 
285 
(tl). = 3, 
{tl)o = 0, 
therefore 3® — 1 numbers have exp 3. 
From these we deduce that 
(3^ — 1)(2^* — 2^®) numbers have exp 3. 2h 
(33_ i)( 213_ oil) „ „ „ 3.23, 
( 33 - 1 ) ( 211 -23) 3 . 2 ^ 
( 33 - 1 ) (23-1) „ „ ,, 3.2. 
Including’ 1 which has exponent 1, this makes the complete set of 2ii. 33 numbers, 
(xxiv.) A special set of generators which generate the h (ni) numbers, modulus m. 
m = {1 iy P/' . . . Qi'"' . . . 
Three numbers (p, (p', (p", generate the residues for modulus (1 + (Pj'opositions 
xiii. and xvi.) Two numbers g^, generate the residues for modulus P/'. (Pro¬ 
position xiia.), &c. One number g\ generates the residues for modulus Qd'. (Pro¬ 
position xii.), &c. 
Suppose any number a modulus m is 
= [mod (1 + U)] 
= (mod P]^^^), 
&c. 
=a\ (mod Qd‘)> 
&c. 
Then 
« = + • • • + “ + • • • (mod m). (Propositiori xix.) 
If now 
rxQ = (p‘°(p'''’’(p'''"'> [mod (1 + ly^ 
=i/iyp(modP/0, 
&c. 
a'l = g'y^ (mod 
&c. 
Thus 
a = ^ + . . . + g'l'^^'i + . . . (mod m) 
= [<l>eo+ ^ 1 + • . . + + • . • + + . • -7“ 
[^''^0 + + • • • + fi + • • -7 “ [^0 + ffi^i + • • • + ^'1 + • ■ -J' 
[^0 +/i^i + • • -7' • • • [^0 + + • • • + + . . -7' • • • (mod m) 
= [(‘^ - 1) ^0 + 17 [( 4>' - 1) ^0 + 1]^'" - 1) ^0 + 17'“ [{(h - 1) + 1]^^ 
[(/i - 1) + 0^' • • • [(7i - 1) + 17'‘ • ' • (mod m). 
