The theory of numbers, by Robert D. Carmichael ...

50

THEORY OF NUMBERS

An interesting proof of this theorem on wholly different
principles may be given. Let p points be distributed at equal
intervals on the circumference of a circle. The whole number
of p-gons which can be formed by joining up these p points
in every possible order is evidently
2I (p-I)(p-2). 3.2.I;
2P
for the first vertex can be chosen in p ways, the second in p -
ways,..., the (p-I)th in two ways, and the last in one
way; and in counting up thus we have evidently counted each
polygon 2p times, once for each vertex and for each direction
from the vertex around the polygon. Of the total number
of polygons -(p-i) are regular (convex or stellated) so that
a revolution through 36o~/p brings each of these into coincidence with its former position. The number of remaining
p-gons must be divisible by p; for with each such p-gon we may
associate the p- p-gons which can be obtained from it by
rotating it through successive angles of 36o~/p. That is,
-p(p- I)(p -2)... 3.2 I-(p- I) mod p.
2p 2
Hence
(P —I)(P -2)... 3-2I-P+I1O mod p;
and from this it follows that
12.... -i+ioo modp,
as was to be proved.
~ 26. THE CONVERSE OF WILSON'S THEOREM
Wilson's theorem is noteworthy in that its converse is also
true. The converse may be stated as follows:
Every integer n such that the congruence
1.23... n-I+I=-O mod
is satisfied is a prime number.