Some Pythagorean type equations concerning arithmetic functions

Abstract

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x 2 + y 2 = z 2 , we study equation f(x 2 ) + f(y 2 ) = f(z 2 ) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning. Subject Classification: 97F60, 11A25