250 
MR. G. T. BERRETT OR” THE RESIDUES OF POWERS OF LUMBERS 
Number of I'esidues prime to any modulus. 
Let the modulus be expressed as a product of powers of its prime, factors, 
m = p'^cf . . . then of the N [m) residues, 
cp ion) = [N (^U) - N (p“-’)] [N {q^) - N . . . 
are prime to m. 
In particular for a pure prime p, 
cp (p) = 'f — I and fp (p“) ; 
and for a mixed prime, a -|- bi, 
{a -j- bi) = rr + Id — 1 and <;P {a + bi) = {<d + b'-f — (a~ + 6*)“ \ 
This value T-, when we are dealing with complex numbers, must be distinguished 
from the value ^ when we are concerned only with real liumbers. 
To a mixed prime (/> is inapplicable, and to a real it has the relations </> (p) = p — 1, 
(f) (p“) z= p“ — p““h 
Any modulus may be midiiplied by — 1, i, or — i, for if the modulus is a factor of 
any number it remains a factor on being multiplied by — 1, i, or — i. This enables 
us to change any modulus into one entirely positive :— 
For 
— a — /3l = — i(fx + /3i) 
— f3 ai = f (a + /3i) 
j3 — ai — — f (a -(- /S<) 
> 
Part IT. —Residues of Powers of Numbers for any Modulus, Composite 
AND Complex. 
(i.) a being a number prime to m each term of the series of residues 
a, a", a®, . . . (mod m) 
is one of the (]> [m) residues which are prime to m. Hence, as in (Proposition l), we 
see that, if t is the smallest integer for which a'' = 1 (mod m), the infinite series of 
residues consists of a repetition of t terms begiiming from the first term. The 
num])er t is called the exponent of a for the modulus m. 
Example .-—The series of residues of the first twenty-four powers of the number 
2 "h f for the modulus 9 are 
