Prime Numbers
Prime Numbers Prime Numbers
3.9 Research problems 171 3.50. The closing theme of the chapter, analytic prime-counting, involves the Riemann zeta function in a certain way. Pursuant to Exercise 1.60, consider the following research path, whereby we use information about the zeta function within, rather than to the right of, the critical strip. Start with the Riemann–von Mangoldt formula, closely reminiscent of (1.23) and involving the π∗ function in (3.25): π ∗ (x) =li0(x) − li0(x ρ ∞ dt ) − ln 2 + x t(t2 − 1) ln t , ρ observing the computational cautions of Exercise 1.36 such as the need to employ Ei for reliable results. The zeros ρ here are the Riemann critical zeros, and one may replace the sum with twice a sum over real parts. The research problem then is: Find a computationally rapid algorithm to estimate π(x) extremely accurately using a collection of Riemann critical zeros. It is known that with a few zeros, say, one may actually compute π(x) as the integer-valued staircase that it is, at least up to some x depending on how many zeros are employed. A hard extension to this problem is then: Given x, how far does one have to go up the critical line with ρ values to compute a numerical approximation—call it πa(x)—in order to have π(n) =⌊πa(n +1/2)⌋ hold exactly for every integer n ∈ [2,x]? We certainly expect on theoretical grounds that one must need at least O( √ x) values of ρ, but the idea here is to have an analytically precise function πa(x) foragiven range on x. References on the use of Riemann critical zeros for prime-counting are [Riesel and Göhl 1970] and [Borwein et al. 2000].
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. ✷
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3.9 Research problems 171<br />
3.50. The closing theme of the chapter, analytic prime-counting, involves<br />
the Riemann zeta function in a certain way. Pursuant to Exercise 1.60,<br />
consider the following research path, whereby we use information about the<br />
zeta function within, rather than to the right of, the critical strip.<br />
Start with the Riemann–von Mangoldt formula, closely reminiscent of<br />
(1.23) and involving the π∗ function in (3.25):<br />
π ∗ (x) =li0(x) − <br />
li0(x ρ ∞<br />
dt<br />
) − ln 2 +<br />
x t(t2 − 1) ln t ,<br />
ρ<br />
observing the computational cautions of Exercise 1.36 such as the need to<br />
employ Ei for reliable results. The zeros ρ here are the Riemann critical zeros,<br />
and one may replace the sum with twice a sum over real parts.<br />
The research problem then is: Find a computationally rapid algorithm<br />
to estimate π(x) extremely accurately using a collection of Riemann critical<br />
zeros. It is known that with a few zeros, say, one may actually compute π(x)<br />
as the integer-valued staircase that it is, at least up to some x depending<br />
on how many zeros are employed. A hard extension to this problem is<br />
then: Given x, how far does one have to go up the critical line with ρ<br />
values to compute a numerical approximation—call it πa(x)—in order to have<br />
π(n) =⌊πa(n +1/2)⌋ hold exactly for every integer n ∈ [2,x]? We certainly<br />
expect on theoretical grounds that one must need at least O( √ x) values of ρ,<br />
but the idea here is to have an analytically precise function πa(x) foragiven<br />
range on x.<br />
References on the use of Riemann critical zeros for prime-counting are<br />
[Riesel and Göhl 1970] and [Borwein et al. 2000].