288 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OE NUMBERS 
a,“ + is a real odd prime number, and therefore 
^ (QdO 
is even, and therefore k'i always occurs. 
For the numbers, Kq k'q k'q, all three occur if 
two occur if 
one occurs if 
none occur if 
K > 4 [m = (l + Pd' • • • Qd' • • •]> 
K= 4, 
k=3,2, 1, 
= 0 . 
Hence, if we denote by n the number of different prime factors, Pj Po. . . Qi Q2 . . . 
of m (excluding 1 + ^) then the number of terms in the set of numbers Kq k'q k'q k^. . . 
k'i .. . is 
n + 3 if K > 4, 
n 2 if K = 4, 
n-\- i if « = 3, 2, 1, 
n if /c = 0. 
These numbers give an inferior limit to the least number of generators. In all but 
one exceptional case the least number of generators coincides with these. 
Consider next the set of numbers, U . . . // V • • . 
// occurs if divides 
cp (Pd‘) = Pd<"'-’HPd- !)• 
If it divides 
Pf - 1 
it occurs once. If it is identical with Pj then it occurs twice (/j = Xj — 1), and the 
set of numbers would be written /, /o . . . U . . , 
occurs if 'p divides 
T (Qd')- 
Therefore the number of terms in the set exceeds n (and is equal to n -f I) 
when m is such that 
T (QP‘), (Q2''^), . . . ^ (PC)’ ^ (P2^') • • • 
are all divisible by p and also one of the primes, P„ P.,, ... is equal to p. 
If this be the case when k — 0 the least number of generators is n 1. 
