206 MR. (4. T. EEXNETT OX THE RESIDUES OE POWERS OF XOIBERS 
So of the three numbers 2* — 1, 2'' ^ + 1,2* ^ — 1, it is always 2'' ^ + 1 to which 
the power 2"'^ of any number with exponent 2'‘"‘ is congruent. 
Let /he either of the two numbers 2'^ — 1, 2'"“^ —1 ; so that p"* * ^/’(mod 2*). 
Consider the 2''“^ numbers 
(mod 2"). 
Clearly no two in the first row are congruent, nor in the second. 
The supposition 
/g’’ = (mod 2/ 
leads to 
ff {/ -f) = h (mod 2'^) 
g^ = f (mod 2''), 
which is contrary to the supposition that g~‘"~^, and therefore no power of g is 
congruent to f. 
Therefore no two of the above 2'' “' odd numbers are congruent mod 2", and hence 
they are the 2* “ ^ numbers (mod 2'^). 
If for y we take 2'' — 1 ^ — 1 (mod 2*), the 2''"^ numbers may be written 
±g, ± g~\ ± . ± " (mod 2''). 
Whichever number be taken for/, any number (mod 2*) is expressible in the form 
a = f ^g^ (mod 2''), 
where 
i is referred to mod 2, '] 
j is referred to mod 2^" “ 
Note. —In the last propositions relating to mod 2'^, 'k is supposed to be > 6. 
In the case k = 2, when the modulus is 2^ = 4, there are two odd numbers, less 
than the modulus, viz., 1 and 3, and 3 (having exponent 2) is a primitive root. 
In the cirse k = 1 when the modulus is 2, the only odd number is unity. 
We see that the case when the modulus is a power of 2, difters very much from the 
case when the modulus is a power of an odd prime. 
When the mod is 2" (k > 3). 
(i.) The highest exponent is not (f) {2") = 2''“\ but 2'"”^, and hence there are 
no primitive roots. 
(ii.) The form of the numbers which belong to aiw exponent is known, and the 
numbers can be at once written down when k is known. In particular 
the numbers ±1 + 2^ (mod 8) always belong to the highest exponent. 
