As we go to press there has been a lot of excitement in mathematics about the proof of the ABC conjecture. The conjecture, which was posed by two prominent number theorists Joseph Oesterlé and David Masser in 1985, is considered to be one of the most important conjectures in number theory and the proof of the conjecture is a notoriously hard problem. It may be that the proof will be revealed to be invalid but the mathematican who came up with the proof, Sinichi Mochizuki, is very highly regarded so it may well be valid. Understandably it will take experts quite some time to wade through the 500 or so pages of proof.
So, what is the ABC conjecture? There are many different ways of stating the conjecture, some easier to understand than others. To begin, you might try the following exercise: take any two positive integers $a$ and $b$ having no common factor and add them to obtain a larger integer $c$; write down all the prime factors of $a,b$ and $c$, ignoring any repetitions; multiply these primes to obtain a product $P$. The aim is to choose $a$ and $b$ in such a way that $c$ is larger than $P$: you will usually find that this is not the case. Here are a few examples.