A Problem in Pythagorean Arithmetic

Abstract

Problem 2 at the 56th International Mathematical Olympiad (2015) asks for all triples $(a,b,c)$ of positive integers for which $ab-c$, $bc-a$, and $ca-b$ are all powers of $2$. We show that this problem requires only a primitive form of arithmetic, going back to the Pythagoreans, which is the arithmetic of the even and the odd.