FOR ANY COMPOSITE MODULUS, REAL OR COMPLEX. 
263 
The square of each of the 8 numbers with exponent 4 gives the same number with 
exponent 2, viz. 7. 
Also 7 = 3. 5 = (1 + 2i) (3 + 2i) = (5 + 2i) (7 + 2i). [mod (1 + 
Hence, firstly, we must take as generators 
and 
one number with exponent 4 
two numbers with exponent 2 
and, secondly, the two numbers with exponent 2 most neither of them be 7, and 
they must not be a pair of the numbers as arranged above whose product is con¬ 
gruent to 7. 
Mod (1 -h 0*"- (1 + 7)® = 32. The 32 numbers with their exponents are 
arranged below. 
Exp 1. Exp 2. 
Exp 4. 
1 5 + 47 
1+27 
1 + 67 
3 + 27 
3 + 6r 
5 + 27 
5 + 6i 
7 + 2'r 7 + 67 
3 + 47 
2 + 7 
2 + 3i 
2 + 57 
2 + 77 
6 + 7 
6 + 3r 
6 + 57 6 + 7r 
7 
3 
5 
1 + 47 
7 + 47 
4 + 7 
4 + 37 
4 + or 
4 + 77 
7 
37 
57 77 
The 24 numbers with exponent 4 are written in three rows, and the number with 
exponent 2, to which the square of any one of them is congruent, is to be found 
written at the end of the row. 
Moreover, the three numbers, with exponent 2 (viz. 5 + 47, 3 + 47 and 7), which 
thus appear as the squares of numbers with exponent 4, are such that the product 
of any two is congruent to the third ; for 
7 (5 + 47) (3 -f 4i) = 1 [mod (1 -f 7)*^]. 
H ence, to generate the 32 numbers, we must take 
and 
two numbers, with exponent 4, not in the same row 
one number, with exponent 2, selected from 3, 5, 1 + 4<, 7 -f 4n 
