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NUMBERS. 
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power which is =1. For instance, say N = 24>, <£(I7) = 8, then the numbers less than 
24 and prime to it are 1, 5, 7, 11, 13, 17, 19, 23; and we have 
l 1 = 1, 5 2 = 7 2 = 13 2 = 17 2 = 19 2 = 23 2 = 1 (mod. 24), 
that is, 1 has the exponent 1, but all the other numbers have the exponent 2. So- 
again where 17 = 48, the 16 numbers less than 48 and prime to it have, 1 the 
exponent 1, and 7, 13, 17, 23, 25, 31, 35, 41, 47 each the exponent 2, and the 
remaining numbers 5, 11, 19, 29, 37, 43 each the exponent 4. We cannot in this case 
by means of any single root or of an}’ two roots express all the numbers, but we 
can by means of three roots, for instance, 5, 7, 13, express all the numbers less than 
48 and prime to it; the numbers are in fact = 5 a 7^13>', where a = 0, 1, 2, or 3, and /3 
and y each =0 or 1. 
Comparing with the theorem for a prime number p, where the several numbers 
1, 2, 3, p—1, are expressed by means of a single prime root, =g a , where a=0,1, 2, 1, 
we have the analogue of a case presenting itself in the theory of quadratic forms,— 
the “irregularity” of a determinant (post, No. 31); the difference is that here (the 
law being known, 17 = a composite number) the case is not regarded as an irregular 
one, while the irregular determinants do not present themselves according to any apparent 
law. 
21. Maximum indicator—application to solution of a linear congruence. In the 
case 17=48 it was seen that the exponents were 1, 2, 4, the largest exponent 4 being 
divisible by each of the others, and this property is a general one, viz. if N =a a №cy... 
in the series of exponents (or, as Cauchy calls them, indicators) of the numbers 
less than N and prime to it, the largest exponent / is a multiple of each of the 
other exponents, and this largest exponent Cauchy calls the maximum indicator; the 
maximum indicator I is thus a submultiple of </>(17), and it is the smallest number 
such that for every number x less than N and prime to it we have x 1 — 1=0 (mod. 17). 
The values of I have been tabulated from N = 2 to 1000. 
Reverting to the linear congruence ax= c (mod. b), where a and b are prime to 
each other, then, if I is the maximum indicator for the modulus b, we have a 1 = 1, 
and hence it at once appears that the solution of the congruence is x = ca z ~ l . 
22. Residues of powers for an odd prime modulus. For the modulus p, if g be 
a prime root, then every number not divisible by p is = one of the series of numbers 
g, g 2 , ..., g p ~ l ; and, if k be any positive number prime to p — 1, then raising each of 
these to the power k we reproduce in a different order the same series of numbers 
g, g 2 , ..., g p ~ l , which numbers are in a different order =1, 2, ..., p— 1, that is, the 
residue of a &th power may be any number whatever of the series 1, 2, ..., p — 1. 
But, if k is not prime to p — 1, say their greatest common measure is e, and 
that we have p — 1 = ef k = me, then for any number not divisible by p the &th power 
is = one of the series of f numbers g e , g 2e , ..., g fe ; there are thus only f = -^ (p — 1), 
out of the p—1 numbers 1, 2, 3, ..., p —1, which are residues of a &th power, 
c. xi. 76