Last week I wrote about the state of modern physics, and how - in my opinion - it was now so complex and filled with subtle technicalities that an untrained amateur would have no chance of producing some great new theory or result. However it is interesting to note that although theoretical physics and mathematics are very closely related, the field of mathematics is still ripe for amateurs to produce important results. Even primary school students can understand mathematics problems that the greatest mathematicians of history cannot answer. 

And so today I present for you, three interesting unsolved problems in number theory.

Most people know what a prime number is - it is an integer (also called a whole number) which is divisible only by itself and the number one. For example, 2,3,5,7, 11, and 13 are all prime numbers, while 4,6,8,9,10, and 12 are not. The study of prime numbers is as old as mathematics itself. 

And so you would think that by now, after centuries of study by the greatest minds in history, we would know everything there is to know about them. And you would be wrong.

Of the many prime number problems which are still unsolved, here are three of the most famous:

1. The Twin Prime Problem: The sample prime numbers given above contain the pairs (3,5), (5,7) and (11,13), each of which is formed of two prime numbers whose difference is 2. Even among the largest prime numbers, there are pairs that obey this same property, with the record being the pair 3756801695685 × 2666669 ± 1 which was discovered just four years ago. However no one knows if there exists a largest pair. As the primes get larger, the average gap between them gets larger, and yet so far there are always a few prime pairs that break the pattern. Do they continue to infinity?

2. The Goldbach Conjecture: In the year 1742 the mathematician Goldbach wrote to the legendary mathematician Euler to discuss an interesting property of prime numbers. He observed that every number could be written as the sum of three prime numbers, which Euler improved to say that every even number can be written as the sum of exactly two prime numbers. This has been tested for all numbers up to 4 x 1018, and yet in the past 270 years no one has ever succeeded in proving that it is true in general. Is there a huge number which requires four prime numbers? Or is the Goldbach Conjecture true for all numbers?

3. The Riemann Hypothesis: The most interesting question of all in number theory is whether there is a general formula for generating prime numbers. While this is a huge subject worthy of a separate article, it seems to reduce to the question of whether the Riemann hypothesis is correct. There are formulae which predict prime numbers, but the error in these is related to a simple sum over the integers, of (1/N)s where s is a complex number. The Riemann hypothesis claims that this sum is only zero when the real part of s is equal to 1/2 (aside from some negative values of Re(s)). As with the previous puzzles, this has been tested with powerful supercomputers for very large numbers and no exception has ever been found. But no one has been able to prove that it is true either. This problem is so famous and so important that whoever solves it will receive over $1,000,000 in prize money and numerous major mathematics awards.

Each of these three problems is accessible to anyone with a basic knowledge of arithmetic, and yet they have never been solved. So perhaps one day, an amateur mathematician from some obscure realm will come forward with the answer.

It is a wonderful thought to realize that mathematics is so powerful, and yet still simple enough to be open to anyone with an interest.