FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
(xv.) The numhers which have exponent 2 '^/or mod (1 + ^)^ [X >8 and 5 > 3 .] 
The numbers are those which satisfy 
= 1 [mod (1 + 
and which do not satisfy 
^ = 1 [mod (1 4 - 
The first gives 
« = d: I 5 i ^ [mod (1 + if X — 25 > 2, 
and the second 
a= d: 1? i 0 [mod (1 iy 2 'S + 3j_ 
Hence the numbers with exponent 2-' for mod (1 -f (X — 25 > 2), are 
d: 1, ± L + (1 + 1 
dz dz b + (1 + + ^ >.mod (1 -f 
dz 1, ziz i, + (1 + + (1 + ^y-'^+y 
-2^+3 
The number of them is 
12 X N (1 
= 2b 3. 22^-2 
— 2-2^ d- 22^ + h 
y mod (1 d- 'if, 
If X is odd, = 2/r d- 1 the greatest value of 5 is /a — I, and then X — 25 > 2. 
So there are 2^^*“^ -f 2^^“^ numbers with exponent 2'^“^ for mod (1 -f i)~'^'^^, viz. 
dz 1, ± i, + (1 d- if 
± 1, ±b + (l 
dz 1, ±b + (l+^)' + (l J 
7 d- 2 l 5 d- 2i, 6 d- 3b 6 -f- ^ 1 
= 5 3 A -\- i 3i > mod (1 
3 d- 2b 1 d- 2b 2 d- 3b 2 d- i . 
+ ^y 
So for mod (1 -b if'^'^^ there are 
22 /^-i _|- 2 ^'" ^ numbers with exp 2'" ^ 
£2^ + 1 _p 2 ^* 
5? 
?5 
2 * 
V + 2» 
>? 
?? 
23 1 
25 d- 2* d- 23 
5 ) 
3 J 
22 
2 '^ + 2 + 1 
> ? 
33 
2 
1 
3 3 
1 
making, in all, 2^'^ = <l) (l numbers, as it should. 
