Written: Oct. 6, 2005.
A few years ago I went to see David Auburn's Broadway play "Proof", and a few days ago I watched the movie. While I really liked both versions (even though the movie's script was made trashier and less subtle for obvious reasons), until almost the end, I was very disappointed about the end in both versions. I don't remember the exact wordings, but the methods that were used in the play's proof used buzzwords from Wiles's proof of FLT. For the movie version, it was `non-commutative geometry and random matrices', obviously an allusion to Connes's approach and the Montgomery-Odlyzko-Conrey-Sarnak-Katz-et-al. approach to RH.
Neither approaches can be plausibly used by someone who only took one semester of math at Northwestern. It would have been much more convincing to have Catherine find an elementary combinatorial proof, using entirely new ideas that escaped the mainstream.
I couldn't help remembering the beautiful ending to my favorite math novel and one of my all-time-favorite books, Apostolos Doxiadis's masterpiece "Uncle Petros and the Goldbach Conjecture". Uncle Petros's approach was completely elementary and combinatorial, using beans! This is exactly the approach introduced by James Joseph Sylvester in 1880 to study partition theory and beautifully exploited by him and his Johns Hopkins students to prove fairly deep theorems, some of which were brand-new at the time, and others for which only analytic proofs were known.
One is also reminded of Viggo Brun's entirely elementary approach to Goldbach, essentially using an extension of inclusion-exclusion. Along with Vinogradov's analytic approach, this remains the most promising avenue.
I also loved the calculated ambiguity at the very end of "Uncle Petros" - the "official" vs. the "unofficial" opinions of the narrator. Contrast this to both versions of "Proof", but especially the movie one, where there is no ambiguity left. Catherine `obviously' was the prover, because her father was very unlikely to know about non-commutative geometry and random matrices (or modular forms and L-functions). QED. This "circumstantial evidence" proof is faulty even in the colloquial or judicial sense of the word, since as I have already mentioned, Catherine was just as unlikely as her father to master these high-brow fashionable mainstream methods and to succeed where the mainstream mafia failed. Her only chance of success would have been to develop an entirely new, preferably elementary and combinatorial proof, with or without beans.
Doron Zeilberger's Homepage