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):=A^{2}x^{2}+ Bxy+ Cy^{2}

Then:

4Q(c-Ca, Ba-Ab)=(2Ac+2ACa-Bb)^{2}
-(B^{2}-4A^{2}C)(b^{2}-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