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

THE THEOREMS OF FERMAT AND WILSON

55.

EXERCISES
I. Show that a16rI mod 16320, for every a which is prime to 16320.
2. Show that a12"I mod 65520, for every a which is prime to 65520.
3*. Find one or more composite numbers P such that
aP-1I mod P
for every a prime to P. (Compare this problem with the next section.)
~ 29. ON THE CONVERSE OF FERMAT'S SIMPLE THEOREM
The fact that the converse of Wilson's theorem is a true
proposition leads one naturally to inquire whether the converse of Fermat's simple theorem is true. Thus, we may ask the
question: Does the existence of the congruence 2n- 1-I mod n
require that n be a prime number? The Chinese answered
this question in the affirmative and the answer passed unchallenged among them for many years. An example is sufficient
to show that the theorem is not true. We shall show that
2340 I mod 341
although 34I,=II'3I, is not a prime number. Now 20-I
=3 II*3I. Hence 21~I mod34I. Hence 2340- mod 34I.
From this it follows that the direct converse of Fermat's theorem is not true. The following theorem, however, which is
a converse with an extended hypothesis, is readily proved.
If there exists an integer a such that
an-l-I mod n
and if further there does not exist an integer v less than n-I such
that
av I mod n,
then the integer n is a prime number.
For, if n is not prime, ~(n)<n-I. Then for v=~(n) we
have av- i mod n, contrary to the hypothesis of the theorem.