P v NP Papers Galore
Computational Complexity 2025-04-30
As someone who has literally written a book on the topic, I have had many people over the years send me their attempts at P v NP proofs. On average, I receive about one a month, but I've had seven in the last two weeks. And not just the usual email with a PDF attachment. A DM in X. A phone call with a baby in the background. Via Linkedin, in Spanish. One with forty-one follow up emails.
P v NP is still the most important problem in theoretical computer science and perhaps all of mathematics, and not that difficult to understand, at least on an intuitive level. I can see the fascination in wanting to solve it, much the way my teenage self dreamed of being the first to prove Fermat's last theorem (damn you Wiles).
But why the spate of "proofs" right now? Maybe it's an artifact of the crazy world we live in.
In one sense, I appreciate the continued interest in this great question, even though these papers rarely (really never) provide new insights into the P v NP problem. Most of my colleagues just ignore these emails, I usually try to respond once, but most authors will come back and I just don't have time for those continued conversations.
These papers come in three categories.
- Claiming to prove P ≠ NP by arguing that a polynomial-time machine must search a large space of solutions. To truly prove P ≠ NP, one cannot make assumptions about how the machine might work.
- Giving an algorithm for an NP-complete problem which works on some small test case. I'd usually ask for them to solve SAT competition problems, solve RSA challenge problems or mine themselves some bitcoin.
- Giving a new philosophical or physical spin on the P v NP problem and claiming that tells us about whether P = NP. But P v NP is at its heart a mathematical question, and no amount of philosophy or physics will help settle it.