202 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
We have 
rjjk+i = 1 (mod 2^^^). 
Therefore 
rj'A+i = I (mod 2^); 
tlierefore 
Again 
Therefore 
and therefore 
divides (Prop. 2.) 
= 1 (mod 2^) = 1 + a;. 2^ + y. 2 ^""^ (where a: = 1 or 0). 
= 1 (mod 2^’^^), 
Therefore either 
4+1 divides 2t^. 
or 
Suppose now that after 
we can have 
Then 
4+2 — 4 + 1 - 
= 1 (mod 2"^) = 1 -j- a’. 2^ -{- 2^’^f 
Therefore 
Now if X>2, then 
and therefore 
Now 
therefore 
Therefore 
rr'A = 1 + a;.2"+* + a;-2''^ (mod 2"+'). 
2 \;\ + 2 , 
= 1 + a;. 2^"^^ (mod 2^'‘‘"). 
cr'A =2 ff ^+2 = 1 (mod 2^'^"), 
X — 0. 
= 1 d- -^.2^ + y2^'^^ = 1 (mod 2^'*'‘), 
thei’efore 
^+1 divides 
which is not so. 
Hence we have tlie resuit, when X = 2 and = 2t)^, then 
The hrst exponent of the series, is equal to unity. 
If this be followed by a set of Ps (at least one), then by what has been proved 
will be followed by the series 2, 2-, 2®.. . . 
