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When would you read a paper claiming to have settled a famous long open problem like $P$ vs. $NP$?

If the paper claimed to resolve $P$ vs $NP$, I'd begin reading it right away. For instance, I'm currently looking at this paper. But that is only because I have a good chance of understanding the work. If the paper claimed to resolve any other Clay Millennium Prize problem, I'd defer to others.

What are your criteria to consider such an announcement as serious?

(a) It's not written in Microsoft Word

(b) The abstract, title, and opening paragraphs do not convey obvious misunderstandings

For $P$ vs $NP$ proofs, criteria (a) and (b) work about 99% of the time. (Seriously.) I'm still checking to see if the above link passes criterion (b). It's a long abstract.