This essay by journalist Caroline Chen describes the ABC conjecture and its story, including the fact that mathematician Shinichi Mochizuki posted a proof not quite a year ago, and the mathematical community has yet to confirm or even understand it.
Three years ago when the P ≠ NP conjecture was purportedly proven, I blogged about it and was excited for the social process of checking the proof. Unfortunately it was pretty quickly decided that there were large holes in the reasoning and it wasn't even a proof, though it did give some new ideas on how to solve the problem.
And then just last week, it was announced that Yitang Zhan proved the "bounded gaps" conjecture for prime numbers. The paper was submitted to the Annals of Mathematics in April, and was accepted 2 days ago, meaning that the proof has passed peer-review.
What will happen with the ABC conjecture? No one has found holes in Mochizuki's proof, but no one can understand it either. My favorite quote from Chen's essay:
"[In mathematics,] Colleagues check each other’s work, spending hours upon hours verifying that a peer got it right. This behavior is not just altruistic, but also necessary: unlike in medical science, where you know you’re right if the patient is cured, or in engineering, where the rocket either launches or it doesn’t, theoretical math, better known as 'pure' math, has no physical, visible standard. It is entirely based on logic. To know you’re right means you need someone else, preferably many other people, to walk in your footsteps and confirm that every step was made on solid ground."