Prime Numbers

152 Chapter 3 RECOGNIZING PRIMES AND COMPOSITES
we have
(−1) S
disc(f)
= .
p
Assertion (1) follows, since Fi(x) is precisely the product of the degree-i
irreducible factors of f(x), so its degree is a multiple of i. Assertion (2) holds
for all polynomials in Fp[x]. Assertion (3) is a little trickier to see. The idea is
to consider the Galois group for the polynomial f(x) overFp. The Frobenius
automorphism (which sends elements of the splitting field of f(x) totheir
p-th powers) of course permutes the roots of f(x) in the splitting field. It acts
as a cyclic permutation of the roots of each irreducible factor, and hence the
sign of the whole permutation is given by −1 to the number of even-degree
irreducible factors. That is, the sign of the Frobenius automorphism is exactly
(−1) S . However, it follows from basic Galois theory that the Galois group of
a polynomial with distinct roots consists solely of even permutations of the
roots if and only if the discriminant of the polynomial is a square. Hence
the sign of the Frobenius automorphism is identical to the Legendre symbol
, which then establishes the third assertion.
disc(f) p
The idea of Grantham is that the above assertions can actually be
numerically checked and done so easily, even if we are not sure that p is prime.
If one of the three assertions does not hold, then p is revealed as composite.
This, then, is the core of the Frobenius test. One says that n is a Frobenius
pseudoprime with respect to the polynomial f(x) ifnis composite, yet the
test does not reveal this.
For many more details, the reader is referred to [Grantham 1998, 2001].
3.7 Counting primes
The prime number theorem (Theorem 1.1.4) predicts approximately the value
of π(x), the number of primes p with p ≤ x. It is interesting to compare these
predictions with actual values, as we did in Section 1.1.5. The computation of
π 10 21 = 21127269486018731928
was certainly not performed by having a computer actually count each and
every prime up to 10 21 . There are far too many of them. So how then was the
task actually accomplished? We give in the next sections two different ways to
approach the interesting problem of prime counting, a combinatorial method
and an analytic method.
3.7.1 Combinatorial method
We shall study here an elegant combinatorial method due to Lagarias, Miller,
and Odlyzko, with roots in the work of Meissel and Lehmer; see [Lagarias et
al. 1985], [Deléglise and Rivat 1996]. The method allows the calculation of
π(x) in bit complexity O x 2/3+ɛ ,usingO x 1/3+ɛ bits of space (memory).