The Twin Prime Conjecture

I was asked a few weeks ago if there are any mathematics problems left that are unsolved, but are sufficiently easy as to be accessible to high school students. Here is an interesting one that is fun to play with and doesn't require advanced math.

The Twin Prime Conjecture has been around for centuries and it remains unproven to this day in spite of being worked on by many high profile mathematicians. At one point it was even the subject of a large cash prize contest to prove it. It is deceptively simple...

Consider the prime numbers. (Quick refresher: A prime number is one that cannot be written as the product of two other integers. For example 5 is a prime number, but 6 = 2x3 is not). The first few are:

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

Notice a pattern? 3 and 5 are only two apart and both are primes. Same for 5,7 and 11,13, and 17,19, and 29,31. They are each only two numbers apart but both are primes. (As an aside, can you see what there can never be two prime numbers only one number apart except for 2,3?)

Modern supercomputers have tested millions of prime numbers of absurd sizes and found this pattern continues. There are runs of tens of thousands of numbers without a single prime and then suddenly there will be a pair of two primes together!

The twin prime conjecture states that this will always be true no matter how large the prime number, but no one has ever been able to prove it. It is an interesting case of a problem that even grade school students understand, but not even the most skilled mathematicians can solve.