SPECIMEN TABLE M^ct a b p (MOD. N) FOB ANY PHIME OB 
COMPOSITE MODULUS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), 
pp. 95—96 and plate.] 
If A be a prime number, and a one of its primitive roots, then any number M 
prime to N, or what is the same thing, any number in the series 1, 2, ...A—1, may 
be exhibited in the form M = a a (Mod. A); where a is said to be the index of M in 
regard to the particular root a. Jacobi’s Canon Anthmeticus (Berlin, 1839), contains a 
series of tables, giving the indices of the numbers 1, 2, 3...A — 1 for every prime 
number N less than 1000, and giving conversely for each such prime number the 
numbers M which correspond to the indices a=l, 2, ... (A - 1) (Tabula} Numerorum ad 
Indices datos pertinentium et Indicum Numero dato cor respondentium). A similar theory 
applies, it is well known, to the composite numbers; the only difference is, that in 
order to exhibit for a given composite number A the different numbers less than A 
and prime to it, we require not a single root a, but two or more roots a, b,... and 
that in terms of these we have M = a a № ... (Mod. A). For each root a there is an 
index A (or say the Indicator of the root), such that a A = 1 (Mod. A), A being the 
least index for which this equation is satisfied; and the indices a, 6,... extend from 
1 to i, if, ... respectively; the number of different combinations or the product AB..., 
being precisely equal to </>(A), the number of integers less than A and prime to it. 
The least common multiple of A, B..., is termed the Maximum Indicator, and repre 
senting it by I, then for any number M not prime to A, we have M 1 = 1 (Mod. A), a 
theorem made use of by Cauchy for the solution of indeterminate equations of the 
first order. Thus A =20, the roots may be taken to be 3, 11; the corresponding 
exponents are 4, 2 (viz. 3 4 = 1 (Mod. 20) 11 2 = 1 (Mod. 20)), and the product of these 
is 8, the number of integers less than 20 and prime to it; the series [go to p. 86] 
11—2