200 MR. G, T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
Again 
= 1 (mod p^), 
therefore (raising to the p*'' power = 1 + Kp^') 
apf\ = 1 (mod 
therefore 
k+\ divides (Prop. 2.) 
It follows from these two results that either or 4+i = P^k' 
Each exponent in the series ... .. . is either equal to that immediately 
preceding it or is p times that value. 
We can show, however, that after the first set of equal exponents t^ = 1^= . . . 
comes to an end, that each exponent is p times that which immediately precedes it. 
For suppose that, if possible, after 
= Pk 
we can have 
We thus have 
^A + 2 - ^A+l- 
Now 
k + -2 — k + \ — Pk- 
= 1 (mod j)^) =1-1- xp^ -\-yp^"" \ 
say, where x<p 
Also, 
therefore 
= f^'A + 2= 1 (modp^'^“), 
(1 -)- xp^ -h yp^ = 1 (mod p^ ■) 
therefore 
1 -p xp^ ^ = 1 (mod 2^^ ") 
X = 0, 
and, therefore. 
r^A = 1 -p yp^ + '■ 
Therefore 
= 1 (mod ^'). 
p + i divides 
which is not so. 
if 
Therefore we cannot have 
^A + 2 — k + l 
^A + 1 = 2^k, 
and therefore we must have 
+ 2 — Pk + 1 
