A problem in Pythagorean Arithmetic

TL;DR

This paper investigates a number theory problem from the 2015 IMO involving triples of positive integers with specific powers of two, revealing that solving it requires only basic Pythagorean arithmetic of even and odd numbers.

Contribution

It demonstrates that the problem can be addressed using elementary Pythagorean arithmetic, simplifying the approach compared to more advanced methods.

Findings

The problem reduces to elementary parity considerations.

Solutions involve only basic properties of even and odd integers.

The approach simplifies the original problem significantly.

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.