190 
MR. G. T. BERNETT ON THE RESIDUES OF POWERS OF NUMBERS 
assist in elucidating the symbolical proofs which they follow; in all cases they w\\\ 
help to maintain clearly the actual arithmetical meaning of the results arrived at, a 
meaning wliich may easily seem obscure if it be noticed only in its symbolical and 
generalised form. 
O 
The A]jpendix contains tal)les of indices for complex numbers for all moduli whose 
norms do not exceed 100. 
PART 1.—ON THE RESIDUES OF POWERS OF NU^fBERS FOR ANY CO^IPOSTTF. 
REAL MODULUS. 
In what follows (except when the contrary is explicitly stated) u'e shall be treating 
(if the residues of powers of numbers which are prime to the modulus with regard to 
which those residues are taken ; and the modulus wall be take]] to be any composite 
]uunbei- whatever. In this first part, ]noreover, all the numbers dealt with are real. 
(1.) The residues (modulus rn) of the successive powers of a number a prime to m 
foiMu a recui'ring series of peilods of tei’ins, the first ]'ie]-iod lieginning with the first 
0 
term. 
( bnsidei' tlie sei'ies of numbers 
a, fr, (r. . . 
Since a is ])rime to v;?, tlierefore any powe]' of k is prime to m, and thei-efoi'e the 
i’esi(bie of a' for modulus ra is ]>rime to m. 
Hence each term of the se]‘ies of insidiies, 
n, O', a'' . . . (mod rn), 
is o]]e of the ]iumbers less than m and prime to it. 
There are (J) (m) numbers less tha]i m aiid prime to it. 
Hence i]i the above infinite seaaes there are only </> ()/]) different terms. 
Su])pose that the first teian which occu]'S fo]’ the second time is (mod m), and 
s]]p]iose tliat this is cong]'iie]it to ot 
Then 
(i< + ^ ^ rP (]nod rn), 
whei’e .s and t are b(Mh to have as s]nall values as possible. 
o' ((P — 1) = 0 (]nod rn), 
a]id since ('P is jiinne to rn. 
(P — 1=0 (mod m), 
a]id so 
(P’’ "■ = d" (]]iod m), 
tbi- every value of a- 
