Posted Feb. 4, 2007.

> Referee's report on > "A Proof of the Loehr-Warrington Amazing TEN to the Power n > Conjecture" > by Shalosh B. Ekhad, Vince Vatter, and Doron > Zeilberger > > > There are several proofs of this conjecture, including some > generalizations, on the internet. > There is an all-human proof by Loehr, Sagan, and Warrington. But more > importantly, there is a > lovely proof by Jonas Sjostrand on the arXiv (already accepted for > publication), > which proves not only the power of ten thing, but also the general > conjecture with a,b replacing 3,5. We admire Jonas Sjostrand's proof very much. It is a real gem! In fact I wrote an expository paper presenting it in an even simpler way, and showing that indeed it is A POSTERIORI trivial, and realizing this fact is indeed a triumph to human ingeniuity. > The method of Zeilberger and Vatter, though very ingenious, > has a fatal flaw. It works only for particular "specific" values of a > and b, > like (3,5). It seems incapable, in principle, of doing the general > case. My > conclusion: in this instance, people defeat computers. YES and NO! Computers lost a battle, but are going to win the war. The implicit methodology of the computer search for a "grammar", and then proving it rigorously should be applicable to many future other problems that will turn out to be A POSTERIORI DEEP, unlike this one. It was just an example to describe a methodology. Analogy: suppose we publish a paper with the proof of 101 x 99=9999 and thereby introduced the general algorithm for long multiplication. Then some young grad student comes up with the ingenious trick (100-1)x(100+1)=100x100-1x1=9999 and "scooped" us, and even more generally proves (a-b)*(a+b)=a^2-b^2. This is a "two-parameter" generalization, but it will not help (very much) with 999868x1987765= ..... > Sometimes such limitations are ok, as when there are no alternatives. But in > this case, with the availability of a fine graph-theoretic proof that > illuminates the true nature of the general problem, The "general" is not that GENERAL, it is just one specific esoteric, cute, but, as it turned out, rather shallow, fact with two parameters, a and b, about a certain set of words. The importance of the METHODOLOGY in our paper far transcends this specific problem. >I don't see why the ERA would want to publish a proof that works only on specific a and b, > which requires us humans to deal with Maple in nontrivial ways in order to > perceive the proof in the first place. BECAUSE the METHODOLOGY of the proof of the result (that can be considered as an example) and the STYLE and the PRESENTATION are ICONIC EXAMPLES of COMPUTER-GENERATED MATH. Also it was the FIRST proof. Also it is still much shorter (for a=3 and b=2) modulo computer computations. Also, the style of the presentation is engaging and not the usual boring style etc. etc.

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