226 MR. G. T. BENNETT ON THE RESIDGES OF POWERS OP NUMBERS 
Hence the number of numbers, with exponent p* generated by the numbers F, is 
and, therefore, we must have 
jpK^'^h _ p\- 
ir 
for all values of s from 1 up to the greatest of the values of the numbers 1. 
Hence wf! get 
(S»)i = (V)i, 
(2«), = (tl)„ 
(Sn), = {Xl)„ 
&c. 
The first of these equations shows that the number of numbers I is the same as that 
of the numbers n. 
The second then shows that, of each set, the number of numbers which exceed 1 is 
the same. 
The third then shows that, of each set, the number of numbers which exceed 2 is 
the same. 
And so on. 
Hence the two sets of numbers l^, Wj, . .. are identical (in some order) 
term for term. 
We have, therefore, shown that the most general set of p-power-exponent 
generators must have as exponents the powers of p, f \ , which occur, one 
each, as a principal factor of ^ (Pf), 4* (^ 3^0 • • • 
When p = 2 we have the special case of the 2-powder-exponent generators. 
First suppose k > 1, then 
2(5k)o- +1_ -1 + ^ — -1, 
and so 
(S«)i + 1 = (2»i)i. 
(Xk), + 1 = (Xn),. 
&c. 
Hence the set of numbers n are identical wdith the set k together with unity: and so 
the exponents of the 2-power-exponent generators are 
2. 2"'. 2''', 2"^ . . . 
Secondly, s'lppose k = 2, then k = 0, and the exponents of the generators are 
2. 2"'. 2"^ 
