Opinion 170: If the American Mathematical Monthly wants to publish proofs of routinley-provable identities, they should at least mention that they are routinely provable

By Doron Zeilberger

Written: March 21, 2019

I have just posted a short note (joint with Shalosh B. Ekhad) critiquing a recent paper, by J. -P. Allouche, that was published in the American Mathematical Monthly. In this article there are lots of humanly-generated proofs, that some people may like, but nowhere is it mentioned that they are all routinely provable by readily available software.

For the benefit of those who do not have the time and patience to read our short note, let us just reproduce the parody at the very end. Since today is Purim, it is appropriate.


Let's use an analogy. Suppose that in the American Mathematical Monthly analog of Egypt in 5000BC there was a papyrus that gave an elegant proof of the following theorem:

Theorem I: 123 x 321 = 39483 ,

and then it commented that it is equivalent to

Theorem II : 111 x 449 = 49839 ,

but that the latter identity is no easier to prove than the former one. In a logical sense, that ancient savant would have been right, since all true statements are logically equivalent. But some mathematical historian could have been tempted to `explain' how Professor Ahmes may have reasoned. Since

111= (123-12) and 449=321 +128 (proofs left to the reader), we have

111 x 449 = (123-12)x(321 +128) = 123x321 + 123x128 - 12x321 - 12x128

=123 x 321 + 15744 - 3582-1536

Using Theorem I, the first product equals 39483, and it follows that indeed

111 x 449 = 39483 + 15744 - 3852-1536 = 49839 QED.

With all due respect, it is much easier to prove Theorems I and II separately, rather than only prove Theorem I, and then deduce Theorem II from it.

Added March 25, 2019: Read Jean-Paul Allouche's interesting response

Added May 18, 2019: Read Christian Krattenthaler's interesting comments

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