Prime Numbers

One of the most important and beautiful fields of mathematics is number theory - the study of numbers and their properties. Despite the fact that mathematicians have been studying numbers for as long as humans have been able to count, the field of number theory is far from being outdated; some of the most exciting and important problems in mathematics today have to do with the study of numbers. In particular, prime numbers are of great interest.

Definition: A number p is prime if it is a positive integer greater than 1 and is divisible by no other positive integers other than 1 and itself.

Positive integers greater than 1 that aren't prime are called composite integers.

Examples: 2,3, and 5 are prime. 6 is composite.

All positive integers n have at least one prime divisor: if n is prime, then it is its own prime divisor. If n is composite, and one factors it completely, one will have reduced n to prime factors.

Examples: 6=3*2, 18=3*3*2, 48=6*8=2*3*2*2*2

The following theorem was proved eloquently by Euclid.

Theorem: There are infinitely many prime numbers.

Proof:

Suppose the opposite, that is, that there are a finite number of prime numbers. Call them p₁, p₂, p₃, p₄,....,p_n. Now consider the number

(p₁*p₂*p₃*...*p_n)+1

Every prime number, when divided into this number, leaves a remainder of one. So this number has no prime factors (remember, by assumption, it's not prime itself). This is a contradiction. Thus there must, in fact, be infinitely many primes.

So, that proves that we'll never find all of the prime numbers because there's an infinite number of them. But that hasn't stopped mathematicians from looking for them, and for asking all kinds of neat questions about prime numbers.