220 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Examples .—The number of numbers that belong to any exponent for mod 
m — 2b 13. 17. 19. 
The greatest exponent is the L.C.M. of 
(f> (2^) = 23 
4>{13) = 2b 3 
4> (17) = 2^ 
(^( 19 ) = 2 . 3 ^ 
> = 2b 3b 
(f) (m) = 210 . 33. 
For the 2-power exponent numbers we have for the ks 
Therefore 
(tK).2 
(2^)3 
(2/c)i 
belonging to exp 2* there are 21° — 2® numbers 
,, ,,23 there are 2® — 2^ numbers 
„ „ 2^ there are 2^ — 2^ numbers j 
,, ,, 2 there are 2® — 1 numbers J 
For the 8-power exponent numbers we may take = 1, ^ = 2, 
(SOi = 2, 
{tl), = 3. 
Belonging to exp 3" there are 33 — 3^ numbers 
„ „ 3 there are 3” — 1 numbers 
The number of numbers belonging to any other exponent is at once got from these 
by multiplying, e.g., the number of numbers with the greatest exponent 2b 3^ is 
(210 — 20) (33 — 3"). 
(24.) We shall now establish a particular set of independent generators which 
genei'ate the </> (m) numbers (mod m) prime to 7n. 
Any number a, mod 711 , is expressible in the form 
a — -h -f + . . . (mod m). 
Let 
