Opinion 132: Bill Duke's Brilliant ("Elementary") Proof of a Seemingly "Deep" Result in Mainstream Number Theory is a Harbinger of Many More Elementary (Computer-Discovered) Proofs to Come

By Doron Zeilberger

Written: May 1, 2013

No one can accuse Bill Duke of being an "experimental mathematican", that alas, at this time of writing, is still, if not a slum, still a "working-class neighborhood" in the metropolis of math. Indeed Bill is one of the greatest ("mainstream") number theorists active today, and as such, I didn't expect to understand much when I went, last Friday, to his talk at the Rutgers Number Theory seminar.

To my pleasant surprise I understood most of it. First, let me mention that Bill is a great speaker! A necessary condition for being a good speaker is only using the blackboard, and writing everything! Even though I am far from an expert, I still followed Bill's talk pretty well (in sharp contrast to the colloquium talk later the same say, that used a laptop, and the speaker spoke at the speed of light, going, in twenty minutes, from D'Alembert to a recent theorem she proved in 2010, when I left with great frustration, getting completely lost).

But the best part was at the very end of Bill's talk! Bill first described some partial results that were proved by heavy-machinery, "deep" Deligne mathematics, and then he concluded with a bombshell! A "trivial" (by hindsight!) polynomial identity that he discovered by "playing around" with Mathematica!

Here it is. Let

Q(x,y):=A2x2+ Bxy+ Cy2


4Q(c-Ca, Ba-Ab)=(2Ac+2ACa-Bb)2 -(B2-4A2C)(b2-4ac)

Voilà Tout!

A much earlier example of a "humble" identity (in this case discovered by hand), but with far-reaching implications to "fancy mathematics", is Bol's identity (Eq. (34) is Marvin Knopp's masterpiece). And I know of a dozen others. But all these were either discovered by hand, or by "playing around" (as in Bill Duke's case above) with a computer algebra system, but still using ad-hoc human exploration.

It is about time to systematize the search for potentially useful identities, and of course, we experimentalists should collaborate with theoreticians, to separate the wheat from the chaff. But don't be too picky, some chaff-looking stuff may turn out to be excellent wheat! It would be a good idea to build a database of potentially useful identities, that who knows, may one day, inter alia, produce a completely "trivial" proof of Fermat's Last Theorem, as fantasized on page 7 of this article of mine.

This project should follow Sara Billey and Bridget Tenner's wonderful manifesto.

Let's get to work!

Added May 9, 2013: In a related vein, see Michael's Somos comprehensive and beautiful compendium of Dededkind Eta Functions identities.
Opinions of Doron Zeilberger