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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