282 
MR. G. T. BENNETT ON THE RESIDUES OF POWERS OF NUMBERS 
With regard to these we shall follow a convention which will be useful in simpli¬ 
fying the next proposition. 
Firstly, we write 2'‘“2'‘'''2''“', and not 2''“Thus the value of (S/c)* will be 
[Kq -1- /Cq'•+ ACy" + ”1- ACg -j- . . . )s, and not (aCq Kq K(^ -f- AC^ -f- + • • • )r 
Secondly, in . . . occurs the prime Pj raised to the power 2 ~ I)- 
We shall suppose it written and not P^* 
For the rest of the princij^al factors no such arrangement is necessary. 
Let 
cl) (Qy^) = . . . 
(Q/") = . . . 
(5rc. &c. 
As in (22) the number of numbers which have some power of p as exponent is the 
product of the number of such numbers for each separate modulus (1 + iy, P/s c&c. 
Now the number of numbers for mod P/‘ is the power of p in (Pi^*) 
P Ao 
>i )5 )) ^ -2 ■>> J5 
O 
J) )> Wl !) ;5 
&c. 
^ (Ps^O >Fi’op. (xxi.). 
Hence the number of numbers with exponent a power of is 
In particular the number of numbers with exponent a power of 2 is 2~‘''h 
(xxiii.) The number of numbers having a given exponent for modulus m, 2/being 
a divisor of the greatest exponent. 
Any such number has a power of p or unity for its exponent for each of the moduli 
(I -f iy, PjN • • ■ &c., and the greatest power of p among these exponents must be p\ 
As in (23) the number of numbers, modulus P/*, which have as exponent a power 
of y> not greater than qf is (if P^ is not p) pF^‘. 
In the particular case when P^ is the prime p, if 
Xj — 1 is > s there are p~^ numbers, mod P^^' with exp a power of p > 
and if 
Xj — 1 is < .S' there are p 
,n2(Ai-l) 
In either case the number is 
Now we have arranged that in T* (P|^‘) the power of P^ {p) in this case) shall be 
written qFpS, and not : thereby we easily express the number of numbers with 
exponent a power of qo p^ for modulus P/', which is or + which 
is not qF^'-'^’. 
For modulus Qj'"’ the number of numbers with exponent a power of jn p^ isp^^'b 
Multiplying these numbers we get as the number of numbers with exponent 
a power of p > qf. 
