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MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
PART II.—ON THE RESIDUES OP POWERS OP NUMBERS FOR ANT MODULUS, 
COMPOSITE AND COMPLEX. 
In order to maintain as far as possible the parallelism of Parts I. and II. a number 
of facts and relations peculiar to complex numbers are noted (most without proof) in 
the following Preface to Part II. 
Preface to Part II. 
Complex lorimes. — i.e., numbers either real or complex which have no real or complex 
factors. 
These are of two kinds 
(i.) Peal primes of the form 4/v + 3 ; e.g. 3, 7, 11, 19, . . . 
(ii.) The factors of real primes of the form 4/j + 1, which are expressible as the 
sum of two squares, e.g., 1 + 2q 2 + -i, 3 + 2i, 2 + 3i . . . Among the last 
we include the factors of the real prime 2, viz., 1 + i. 
These two kinds of prime we shall call respectively^wre and mixed, speaking of either 
as a complex prime. By a real prime we shall mean the primes ordinarily so called, 
real numbers which have no real factors. 
To ex 2 oress omy number cis a product of its p)rime factors. 
Let a + hi be the number. Let d be the G.C.M. of a and h, and sujjpose that 
ct 'T' hi — d (a -|- /3z). 
d is a real number and can be expressed as a product of real primes. 
Of these, those that are not complex primes can be separated each into its two 
factors. 
All the factors of od + fT' are themselves expressible as the sum of tw'o squares. 
Say 
a- + {ap + y8/) . . . 
Then the prime factors of a + /Si are 
it 
± i^2, • • • 
where, in each pair, tlie sign must be so chosen that 
a -j- = (ai d; ?,'y8i) (a^ q- . . . 
Example. 
60 + 105i = 15 (4 + 7i) 
15 = 3. 5 = 3(2 + 0 (2-0 
4^ + 7- = 65 = 5. 13 = (2^ + P) (3^ + 2~) 
