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Currently P vs NP is the holy grail of theoretical computer science. And the nature of the problem is as such that if actually P = NP is proved then most of the proofs for mathematical statements would be trivial to find, and on the exact other end if P != NP is proved then one would appreciate the hardship of finding its own proof finally....

People working in this field are so tired and exhausted of trying to find the solution that many have accepted that they might not even be alive by the time its solved, and many believe we would have to make some other groundbreaking discoveries in math to even be able to think of a rigorous solution.

In fact almost every proof method frequently used to separate complexity classes - like in halting problem's proof - have been proven "insufficient" to solve the problem...

Regardless of the answer being positive or negative, is it possible that if a proof is announced and its actually correct - ONLY for us to realise how trivial it actually was...? like we actually never needed any new concepts... the whole problem was basically hiding right in front of our sight. And maybe or maybe not we require other monumental maths to recognise the trivialness of the problem, but the proof itself, a rigorous and correct one, only needs concepts we have today.....

I know such extreme cases are rare or maybe haven't even occurred yet in maths, but the shear simplicity of the problem - in the sense that a 7th standard student can grasp the essence of the problem efficiently and effectively, just makes me wonder about this situation many a times....

JimN
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Aditya Mishra
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    Of course it is possible (in the same sense that it is possible that someone sends me a correct proof of P=NP by post tomorrow), but I'm not sure how much can be said beyond that. I think this is not so much a question about P vs NP, but rather about philosophy/history of science/mathematics. Or is there anything specific about P vs NP that informs this question beyond it being well-known, unresolved, and relatively simple? – Discrete lizard Nov 07 '23 at 15:24
  • @Discretelizard well, i just can't help but think about this situation evry while, one thing that makes it stand out than most other open problems, is that its actually very easy to understand, basically we give examples of puzzles and games to explain it..... this stands out for me.. although i admit there's very low probability of such a thing happening – Aditya Mishra Nov 07 '23 at 15:31
  • You should check https://cs.stackexchange.com/questions/1877/how-not-to-solve-p-np – Nathaniel Nov 07 '23 at 15:58
  • Fermat thought he solved Fermat's Last Theorem $x^n + y^n = z^n, n \geq 3$, believing the problem and the solution is elementary. Indeed, the problem is elementary. But the solution failed upon scrutiny, and it turned to be one of the most difficult we have ever solved. – Kenneth Kho Feb 12 '24 at 12:04

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Yes of course it is possible. Everything is possible. Such situations have been rare in mathematics, so I would not bet on it, but we have no way to rule out such a possibility. We do know that most of the standard proof techniques we use every day in complexity theory are unlikely to work, which might be taken as some kind of evidence that it's less likely that the proof will turn out to be simple, but you never know.

In any case, it doesn't really have any actionable implications that I can see, so it's probably not worth worrying about too much.

D.W.
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  • I completely agree and think that we have evidences that a proof of $P \neq NP$ will be a very weird one (or maybe we just need to prove $P = NP$ with an algorithm though I doubt it will be an easy one if it exists, the diversity of NP-complete problem makes a case). But I'd like to add that sthg like this actually happened recently with the sensitivity conjecture which stood open for a long time and was solved by Huang in 2 pages with an elementary proof (see here for coverage https://www.quantamagazine.org/mathematician-solves-computer-science-conjecture-in-two-pages-20190725/). – holf Nov 07 '23 at 19:12
  • @holf wow... i do not know that conjecture as of yet , but this is exactly the kind of situation i am talking about... – Aditya Mishra Nov 09 '23 at 14:19
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Consider what happens if I find a polynomial time solution for an NP complete problem. Say a solution taking $n^{100}$ nanoseconds on my computer.

For n = 2 that’s $2^{100} \approx 10^{30}$ nanoseconds. Or $10^{21}$ seconds. Or 30 trillion years. So all but the smallest instance are not trivial, but practically unsolvable.

Now here is a problem that was believed to be very hard and is now known to be trivial: I’ll give you one integer n >= 0 and m >= 1, and you tell me if there are a, b, c >= 1 with at most $10^m$ digits so that $a^n + b^n = c^n$. Finding the answer for n = 3 or n = 4 alone was very, very hard. Today I know that the answer is “Yes” if n=1 or n=2 and “No” otherwise.

gnasher729
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  • yes, the fermat's last theorem, gives a contrary eg to my question, its simple to understand, maybe simpler than p vs np, yet the proof we have as of now is in no way trivial...... – Aditya Mishra Nov 09 '23 at 14:18