Prime Numbers

132 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
Proofs of Fermat’s little theorem may be found in any elementary number
theory text. One particularly easy proof uses induction on a and the binomial
theorem to expand (a +1) n .
When a is coprime to n we may divide both sides of (3.2) by a to obtain
a n−1 ≡ 1(modn). (3.3)
Thus, (3.3) holds whenever n is prime and n does not divide a.
We say that a composite number n is a (Fermat) pseudoprime if (3.2)
holds. For example, n = 91 is a pseudoprime base 3, since 91 is composite
and 3 91 ≡ 3 (mod 91). Similarly, 341 is a pseudoprime base 2. The base a =1
is uninteresting, since every composite number is a pseudoprime base 1. We
suppose now that a ≥ 2.
Theorem 3.4.2. For each fixed integer a ≥ 2, the number of Fermat
pseudoprimes base a that are less than or equal to x is o(π(x)) as x →∞.
That is, Fermat pseudoprimes are rare compared with primes.
For pseudoprimes defined via the congruence (3.3), this theorem was first
proved in [Erdős 1950]. For the possibly larger class of pseudoprimes defined
via (3.2), the theorem was first proved in [Li 1997].
Theorem 3.4.2 tells us that using the Fermat congruence to distinguish
between primes and composites is potentially very useful. However, this was
known as a practical matter long before the Erdős proof.
Note that odd numbers n satisfy (3.3) for a = n−1, so that the congruence
does not say very much about n in this case. If (3.3) holds for a pair n, a,where
1