Prime Numbers

Chapter 4
PRIMALITY PROVING
In Chapter 3 we discussed probabilistic methods for quickly recognizing
composite numbers. If a number is not declared composite by such a test,
it is either prime, or we have been unlucky in our attempt to prove the
number composite. Since we do not expect to witness inordinate strings of
bad luck, after a while we become convinced that the number is prime. We
do not, however, have a proof; rather, we have a conjecture substantiated by
numerical experiments. This chapter is devoted to the topic of how one might
actually prove that a number is prime. Note that primality proving via elliptic
curves is discussed in Section 7.6.
4.1 The n − 1 test
Small numbers can be tested for primality by trial division, but for larger
numbers there are better methods (10 12 is a possible size threshold, but
this depends on the specific computing machinery used). One of these
better methods is based on the same theorem as the simplest of all of
the pseudoprimality tests, namely, Fermat’s little theorem (Theorem 3.4.1).
Known as the n − 1 test, the method somewhat surprisingly suggests that we
try our hand at factoring not n, but n − 1.
4.1.1 The Lucas theorem and Pepin test
We begin with an idea of E. Lucas, from 1876.
Theorem 4.1.1 (Lucas theorem). If a, n are integers with n>1, and
a n−1 ≡ 1(modn), buta (n−1)/q ≡ 1(modn) for every prime q|n − 1, (4.1)
then n is prime.
Proof. The first condition in (4.1) implies that the order of a in Z ∗ n is a
divisor of n − 1, while the second condition implies that the order of a is not
a proper divisor of n − 1; that is, it is equal to n − 1. But the order of a is also
a divisor of ϕ(n), by the Euler theorem (see (2.2)), so n − 1 ≤ ϕ(n). But if
n is composite and has the prime factor p, then both p and n are integers in
{1, 2,...,n} that are not coprime to n, so from the definition of Euler’s totient
function ϕ(n), we have ϕ(n) ≤ n − 2. This is incompatible with n − 1 ≤ ϕ(n),
so it must be the case that n is prime. ✷