Blum’s new paper does not claim to prove P≠NP, but gives his thoughts on P≠NP. I think he has earned our attention. It must have been difficult to go from thinking you have solved the problem to retracting your paper. I have thought, privately, that I had solved some neat problems. Only later to discover that I was wrong. I cannot imagine how tough it was to do this in public.

More here.

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Yes, Norbert Blum has added this comment on the arXiv:

The proof is wrong. I shall elaborate precisely what the mistake is. For doing this, I need some time. I shall put the explanation on my homepage.

So this story is done… for now! I hope everyone trying to prove P = NP is now aware of the Tardos function, the example which could have let Norbert Blum know his proof could not possibly work. There’s now a Wikipedia entry on it:

• Wikipeda, Tardos function.

On G+, Timothy Gowers wrote:

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The difference between Norbert Blum and a crankOne obvious difference is that Blum has already established himself as an extremely reputable researcher: he has published serious results in the area and is very well acquainted with all the literature about techniques that cannot work to prove that P≠NP, giving careful explanations of why they did not apply to his proof attempt. But even stronger evidence of non-crankdom is that when people pointed out that his proof couldn’t work, instead of clinging to it, he retracted it, and now promises to write a detailed account of what is wrong. Similar classy behaviour was exhibited by Edward Nelson a few years ago when his attempted proof of the inconsistency of ZFC turned out to contain a somewhat subtle error.

vloodin wrote:

Tardos’ example seems to refute the proof.

By the way, there’s now a Wikipedia article on this graph invariant introduced by Éva Tardos in 1988:

• Wikipedia, Tardos function.

]]>This newer blog post (by well-known complexity theorists) is well worth reading, even now that the Blum proof appears to have been invalidated: https://rjlipton.wordpress.com/2017/08/17/on-the-edge-of-eclipses-and-pnp/

]]>Thanks! Yes, Razborov is a god of complexity theory, so this report by Mikhail if true is enough for me:

I am familiar with Alexander Razborov whose previous work is extremely crucial and serves as a foundation for Blum’s proof. I had the good luck of meeting him today and wasted no time in asking for his opinion on this whole matter, on whether he had even seen the proof or not and what are his thoughts about it if he did.

To my surprise, he replied that he indeed was aware of Blum’s paper but didn’t care to read it initially. But as more fame was given to it, he did get a chance to read it and detected a flaw immediately: namely that the reasonings given by Berg and Ulfberg hold perfectly for the function of Tardos, and since this is so, Blum’s proof is necessarily incorrect as it contradicts the core of Theorem 6 in his paper.

Idolvon’s remark pointing out in detail “the single error” is also worth reading, though I’m not technically qualified to evaluate it.

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