What is it like to prove a theorem no one has ever proven before?

It’s a special, peculiar, lovely, lonely thrill.

How big of a thrill depends on the circumstances. The theorem could be tiny or huge, famous or unknown, profound or superficial, nifty or stunning, a small improvement over the current state of the art or a complete breakthrough. Just how big of a thrill it can be I can’t tell you: I did not prove anything close to a famous conjecture. But I did get a chance to prove new things, some (I think) interesting and non-trivial. And it was awesome.

Problem solving is great fun (for some of us), but there’s an extra level of satisfaction in solving something that no one solved before – even if perhaps no one even tried to solve it before, but of course even more so if people did.

When you wrestle with a math olympiad or a Putnam problem, it’s like a game: you know that the problem was designed to have a solution. It takes creativity and cleverness and experience to find it, but everyone knows it’s there. But when you wrestle with a conjecture you came up with yourself, or your thesis advisor did, or something that is being investigated by others, then you don’t know if it is at all possible to prove it, or even if it’s true. It’s a mystery, and you feel like exploring a totally new domain: a place no one visited before you.

Finding the final spark that makes everything work is a fantastically satisfying – and rare – pleasure. It may have taken two days, or six weeks, or a few years, and you may have given up on it, and it came back to haunt you, and you gave it more love, and then it paid back.

I don’t know that there are great analogies to explain what it is “like”; perhaps it’s like reaching a summit, but rather than a deliberate expedition to attempt Nanga Parbat, it’s like walking for ages in empty deserts and dense jungles and suddenly finding a brand new peak, and negotiating it, and besting it. Really, I don’t think it’s like anything else.

I wish I possessed the eloquence to describe it.