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

THE THEOREMS OF FERMAT AND WILSON

51

For, if n is not prime, there is some divisor d of n different
from i and less than n. For such a d we have 1.23..
n-i-=omodd; so that 1I2... n-I+ Iomod d; and
hence I 2.... n- + i Go mod n. Since this contradicts our
hypothesis the truth of the theorem follows.
Wilson's theorem and its converse may be combined into
the following elegant theorem:
A necessary and sufficient condition that an integer n is prime
is that
I 2- 3.... n-i +I I mod n.
Theoretically this furnishes a complete and elegant test
as to whether a given number is prime. But, practically,
the labor of applying it is so great that it is useless for verifying
large primes.
~ 27. IMPOSSIBILITY OF I-2 3... n-I+I=-n FOR n>5.
In this section we shall prove the following theorem:
There exists no integer k for which the equation
i- 23...n-I+I=n
is true when n is greater than 5.
If n contains a divisor d different from i and n, the equation is obviously false; for the second member is divisible
by d while the first is not. Hence we need to prove the theorem
only for primes n.
Transposing I to the second member and dividing by n-i
we have
1.2.3... n-2=nk-l +nt-2+... +nn+I.
If n>5 the product on the left contains both the factor 2 and
the factor ~(n-I); that is, the first member contains the factor n-i. But the second member does not contain this factor, since for n=I the expression n'1-+...+n+I is equal
to koo. Hence the theorem follows at once.