While attending a conference in Russia, I went to an art museum with two other mathematicians, and we saw the Bogdanov-Belsky painting. After we did the mental arithmetic depicted, we wondered whether there are infinitely many integers that can be written as both the sum of two consecutive squares and the sum of three consecutive squares. We also discussed the idea of mental math as a kind of gold standard of understanding, at the apex of a hierarchy that has computer-generated proof at the bottom. This led us to conceive of a "math verite" analogous to cinema verite, and we decided to write an article about the Diophantine problem we'd been wondering about, governed by severe strictures analogous to Lars von Trier's "Dogme 95": the article would be an exact transcript of a conversation in which we talked through the problem together without writing anything down. One of the mathematicians in the group, Julian West, had a camcorder, which we used to record our discussion. If I recall correctly, we reduced the problem to a Pellian and then appealed to known facts about Pellians, without actually drilling all the way down to the recurrence relation governing the infinite sequence of solutions. We never followed through on our intention of publishing our "Dogma 2000" article in the Mathematical Intelligencer. In any case the result is far from new (see for instance https://math.stackexchange.com/questions/952216/numbers-represented-as-two-d... as well as oeis.org/A007667), and a discussion of the problem can even be found in one of Martin Gardner's books (see page 22 of "Time Travel and Other Mathematical Bewilderments"). I still like the idea of spoken-word-only math. About a decade ago I came up with a "Bedtime Theorem" proof of V-E+F=2 that the high schoolers I tried it out on, late at night, seemed to follow. But it was interactive, semi-Socratic math, so a transcript (or audio or video capture) wouldn't have the same feel. Jim Propp On Monday, January 1, 2018, Andres Valloud < avalloud@smalltalk.comcastbiz.net> wrote:
Did you get that from Recreative Algebra by Yakov Isidorovich Perelman? His books are full of wonderful problems like this one.
On 12/31/17 18:14 , James Propp wrote:
Here's a (somewhat late) Diophantine puzzle in honor of the 365th day of the year:
It is not hard to show that 365 can be written as both 10^2 + 11^2 + 12^2 and 13^2 + 14^2. (This fact came to my attention about twenty years ago through Nikolai Petrovich Bogdanov-Belsky's painting "Mental Calculation in the Public School Of S. A. Rachinsky", which you can view at https://goo.gl/images/yo88LY.
Puzzle: Show that there are infinitely many integers that can be written both as a sum of two consecutive squares and as a sum of three consecutive squares.
For extra credit, find the proof mentally. :-)
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