31 Dec
2017
31 Dec
'17
7:14 p.m.
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. :-) Jim Propp