795] 
NUMBERS. 
599 
these in any manner so as to form a heptagon; the whole number of heptagons is 
¿.I.2.3.4.5.6. Now of these we have ^(7 — 1), =3, which are regular heptagons 
(convex or stellated); the number of remaining heptagons must be divisible by 7, for 
with any one such heptagon we connect the 6 heptagons which can be obtained from it by 
making it rotate through successive angles of f360°. That is, 1.2.3.4.5.6 —1(7 —1) = 0 
(mod. 7), whence 1.2.3.4.5.6 — 7 + 1 = 0, or finally 1.2.3.4.5.6 + 1 = 0 (mod. 7). It 
is clear that the proof applies without alteration to the case of any prime number p. 
If p is not a prime number, then 1.2.3.1=0 (mod. p); hence the theorem 
shows directly whether a number p is or is not a prime number; but it is not of 
any practical utility for this purpose. 
16. Prime roots of a prime number—application to the binomial equation xP—Y=0. 
Take, for instance, p = 7. By what precedes, we have 
x 3 — 1 = [x 3 — 1] [¿c 3 — 1] [x 2 — 1] [x — 1], = {x 2 — x + 1) (x 2 + x + 1) (x + 1) (x — 1) ; 
and we have 
x 6 — 1 = (x — 1) {x — 2) (x — 3) (x — 4) (x — 5) (x — 6) (mod. 7) ; 
whence also 
(x 2 — x + 1) (x 2 + x + 1) (x + l)(x—l) = (x — 1) (x — 2)(x — 3)(x — 4)(x — 5) (x — 6). 
These two decompositions must agree together, and we in fact have 
x 2 — x + 1 = (x — 3) (x — 5), x 2 + x + 1 = (x — 2) (x — 3), x + 1 = x — 6, x — 1 = x — 1. 
In particular, we thus have 3, 5, as the roots of the congruence x 2 — x + 1 = 0, that is, 
[¿c 6 - 1] = 0, and these roots 3, 5, are not the roots of any other of the congruences 
[ic 3 —1] = 0, [x 2 — 1] = 0, [x —1] = 0; that is, writing a = 3 or 5 in the series of numbers 
a, a 2 , a 3 , a 4 , a 5 , a 6 , we have a 6 as the first term which is = 1 (mod. 7); the series in 
fact are 
3, 9 , 27 , 81 , 243 , 729 =3, 2, 6, 4, 5, 1, 
5, 25, 125, 625, 3125, 15625 = 5, 4, 6, 2, 3, 1. 
And so, in general, the congruence x p ~ x — 1 = 0 (mod. p) has the p — 1 real roots 
1, 2, 3, ...,p — 1; hence the congruence [x^- 1 - 1] = 0, which is of the order (f> (p - 1), has 
this number cf) (p -1) of real roots; and, calling any one of these g, then in the series 
of powers g, g 2 , g 3 , ..., g p ~ x , the first term which is = 1 (mod. p) is g p ~\ that is, we have 
g, g 2 , g 3 , ..., g p ~ 1= = 1, 2, 3, ..., p — 1 in a different order. Any such number g is said 
to be a prime root of p, and the number of prime roots is </> (p — 1), the number of 
integers less than and prime to p — 1. 
The notion of a prime root was applied by Gauss to the solution of the binomial 
equation xP —1=0, or, what is the same thing, to the question of the division of 
the circle (Kreistheilung), see Equation, Nos. 30 and 31, [786] ; and, as remarked in 
the introduction to the present article, the roots or periods of roots of this equation 
present themselves as the units of a complex theory in the Theory of Numbers.