Good question. It's one mathematicians are still trying to answer. The simplest
method was developed by Eratosthenes in the 3rd century B.C. Here's how it works:
Suppose we want to find all the prime numbers between 1 and 64. We write out a table
of these numbers, and proceed as follows. 2 is the first integer greater than 1, so
it is obviously prime. We now cross out all multiples of two. The next number that we
haven't crossed out is 3. We circle it and cross out all its multiples. The next
non-crossed-out number is 5, so we circle it and cross out all its multiples. We only
have to do this for all numbers less than the square root of our upper limit (in this
case sqrt(64)=8) since any composite number in the table must have at least one factor
less than the square root of the upper limit. What's left after this process of
elimination is all the prime numbers between 1 and 64.
Unfortunately, this method is rather time-consuming when the numbers you are looking
for are much larger.
More Prime Number Theory
Mathematicians have developed a great amount of theory concerning prime numbers. Here's
a taste of it:
Question: how far are prime numbers from each other?
Sometimes, only 2 integers apart, like 41 and 43. Although there is a lot of evidence
to suggest it, no one has proved that there are an infinite number of "twin primes."
In general, however, primes get more spread out as they get larger. By how much? In
1896 Charles de la Vallee-Poussin and Jacques Hadamard proved the Prime Number
Theorem, which states:
Let Pr(x) be the number of prime numbers less than x. Then the ratio of Pr(x) to
(x/ln(x)) approaches 1 as x grows without bound.
What this implies is that if n is a prime number, the distance to the next prime number
is, on average, approximately ln(n).
The Goldbach Conjecture
In a letter to
Leonard Euler in 1742,
Christian Goldbach conjectured that every positive even integer greater
than 2 can be written as the sum of two primes. Though computers have verified this up to
a million, no proof has been given.
Since Goldbach's time, however, his idea has been broken down into the 'strong' Goldbach
conjecture - his original claim - and the 'weak' Goldbach conjecture, which claims that every
odd number greater than 7 can be expressed as the sum of three odd primes. Try it and see!