FOR A^Y COMPOSITE MODULUS, REAL OR COMPLEX. 
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and which do not satisfy 
1 [mod (1 + f)^], 
are the numbers with exponent 4. 
Hence Lemma (1) gives us 
ih 1 [mod (1 + 
and then Lemma (2) gives 
a= ih 1, -^i [mod (1 + 
If of these numbers we exclude 
a = dr 1 [mod (1 + iY ~ 
we have the numbers with exponent 4. 
Hence the numbers with exponent 4 are 
and 
The number of these is 
= db [mod (l d" % 
= dzi + (i + ^y-^ 
±i + (id-^y-^ 
> mod (1 + iY~‘^- 
j-l + (l4.,y-4_p(i_p,y-3j 
2N[(l + ^r] + 6N[(l + f)2] 
= 2 . 2 ^ 4 - 2 . 3 . 2 ^ 
= 25 d- 3.23 
= 25 d- 2^ d- 2^ = 56. 
Numbers with exponent 8 mod (1 d“ 'YY' ^>8. 
Tlie numbers are those that satisfy 
(d = 1 [mod (1 d“ ■2')^], 
excluding those that satisfy 
= I [mod (1 d" 
Now 
a® = 1 [mod (1 d" 
gives 
cd ~ ^ \ [mod (1 d“ Lemma (l). 
dr 1 [mod (1 -f- Lemmas (l), (2), and (3) 
a = diL dr [mod (1 -}* Lemmas (1) and (2) 
Of these we have to exclude 
a = dr 1 i b [mod (1 d" 
2 M 2 
