230 MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
(28.) In what follows we shall need the following lemma. 
Consider 
+ \y . . . = hi (mod |P) 
cIqX -|~ ^■'<11 d" -j" • ■ • — ^'2 (mod p ) 
cupc + 632 / + C 32 ; + . . . = I ’3 (mod 
as many congruences as unknown quantities. 
What is necessary in order that the congruences may have one solution, for any 
assigned set of values of the /.’’s, and one only ? 
Multiply the equations in order by the minors of the elements of the 1 st column of 
the determinant 
cq hi Cl 
tto 63 C 3 
and add. 
We get 
• 
Cl . 
a.-, ho Co . 
ko ho Co ' 
— 
^3 ^3 
^3 ^3 
• • 
. 
(mod 
In order that this may have one solution, and one only, the determinant 
' cii hi Cl 
a 3 h^ c.y 
cq &3 C3 
must be prime to and, therefore, prime to pj. 
This being so, then to each set of values of the Fs corresponds a single set of values 
of X, y, z. . . . 
(29.) We have now to find the most general type of a set of j^-power-exponent 
generators. (The work is in no respect ditterent in the case when p = 2 .) 
Let the exponents necessary for a set of generatoi’s be p’h • • • as determined 
by Proposition. (26). 
Let 
gi with exp. p''* "1 
’ ’ [=■ be a set of generators. (Proposition 27.) 
