Opinion 123: Shame On You, the Proceedings of the National Academy of Science (PNAS), for Rejecting a Historic Submission of a Counterexample to Fermat's Last "Theorem"

By Doron Zeilberger

Written: April 1 (!), 2012

I have often protested, in this column (e.g. here, and here), about the narrow-minded stupidity of editors and referees, and because of that I often get Email from many people sharing with me similar experiences. Because of the volume of this correspondence, I am unable to publicize it all (and often the offended party wishes to stay anonymous), but yesterday I got the most shocking example of editorial stupidity and narrow-mindedness, revealed to me by a young computer scientist (who wishes to stay anonymous for now, while his appeal is still pending), that a recent submission to the PNAS, announcing a counterexample to Fermat's Last Theorem was rejected by the PNAS. Here is the abstract.

I know, dear reader, that you are skeptical, and don't believe in this proposed counterexample. After all, the Annals of Mathematics is a very reliable journal, and there is no way that they published a flawed paper. I was sure that the reason for the rejection was the editor's and referee's skepticism. Of course, they know that even the Annals of Mathematics sometimes publishes flawed proofs, e.g. one by ex-mathematician and present-politician D. Biss. But what really shocked me was that both the editor and the PNAS referee do not dispute the veracity of the counterexample. They just claim that it is "not important enough for the PNAS" and should be published in a "specialized journal".

The author kindly allowed me to post the stupid referee report (with the author's name removed).

Being conscientious, and having the greatest respect for Sir Andrew, who in spite of his error is still a mathematical giant, the author sent the counterexample to Andrew Wiles himself, and the latter conceded that he overlooked one case in his proof, where the phrase "it is easily seen ..." turned out to be false as stated. Luckily, Wiles was able to modify his proof to prove a modified statement, that there are at most three counterexamples to FLT. A corrigendum has been submitted to the Annals of Mathematics, and will be published shortly. This still leaves the great computational challenge to find the two other counterexamples (or prove that there is only one). These two even larger counterexamples may be too big to be ever found out, even with twenty websites and using compact notation.

But these are only details. In my humble opinion, a counterexample to the most famous open problem for more than 300 years, that was believed proved for the last seventeen years, is a major breakthrough well worth publication in the PNAS, even though it is "only computational". Shame on you PNAS! How snotty can you get with your conceptual elitism (and I suspect, quite a bit of human chauvinism and machinophobia, because of the heavy use of computers).

Added April 2, 2012: read Drew Sill's suggestion.
Opinions of Doron Zeilberger