Twin Prime Problem

What is a prime number?
Hopefully everyone knows what a prime number is. But for those who don't :
Every number can be written as a product of numbers. For example 10 = 1 x 10 = 2 x 5 . For most numbers this is not unique  - there are many sets of numbers whose product forms our original number - but a few numbers can only be written as themselves times 1. For example, 13=1 x 13 and 29 = 1 x 29 and no other pair of numbers has these products. These numbers are called primes.

So what is the twin prime problem?

If you write down the prime numbers,

2 , 3, 5, 7, 11, 13, 17, 19, 23, 29,31, ...

You notice that they often appear in pairs

2 ,( 3, [5), 7 ],( 11, 13), (17, 19), 23, (29,31), ...

This is mildly interesting for the first primes. However numerical searches for primes have revealed that these pairs continue to very large primes. There are ranges of 1 million numbers which contain no primes, followed by two primes in the next three numbers!

It is believed that these pairs continue to infinity. However no one has ever been able to prove this....