Chapter 1
Elementary Prime
Number Theory, I
Prime numbers are more than any assigned multitude of
prime numbers. Euclid
No prime minister is a prime number A. Plantinga
1. Introduction
Recall that a natural number larger than 1 is called prime if its only positive
divisors are 1 and itself, and composite otherwise. The sequence of primes
begins
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,...
Few topics in number theory attract more attention, popular or professional,
than the theory of prime numbers. It is not hard to see why. The study
of the distribution of the primes possesses in abundance the very features
that draw so many of us to mathematics in the first place: intrinsic beauty,
accessible points of entry, and a lingering sense of mystery embodied in
numerous unpretentious but infuriatingly obstinate open problems.
Put
π(x) := #{p x : p prime}.
Prime number theory begins with the following famous theorem from antiq-
uity:
Theorem 1.1. There are infinitely many primes, i.e., π(x) as x ∞.
1
http://dx.doi.org/10.1090/mbk/068/01
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