The Sequence of Prime Numbers

This section is concerned with three questions:
  1. Are there infinitely many primes?
  2. Given $ a, b\in \mathbb {Z}$ , are there infinitely many primes of the form $ ax+b$ ?
  3. How are the primes spaced along the number line?
We first show that there are infinitely many primes, then state Dirichlet's theorem that if $ \gcd(a,b)=1$ , then $ ax+b$ is a prime for infinitely many values of $ x$ . Finally, we discuss the Prime Number Theorem which asserts that there are asymptotically $ x/\log(x)$ primes less than $ x$ , and we make a connection between this asymptotic formula and the Riemann Hypothesis.


William 2007-06-01