On an Erdős–Straus type problem

Abstract

Motivated by the Erdős–Straus conjecture, we study whether the Euler totient ratio φ(n)/n can be expressed as a sum of three unit fractions,

φ(n)/n=1/x+1/y+1/z,       x, y, z ∈ N.

We prove that such a representation is impossible for prime powers p^k with p ≥ 11, and for integers of the form p^k1q^k2 with p ≥ 5 and q ≥ 15p. In contrast, for 2 ≤ p ≤ 7 the representation exists for all p^k. Moreover, for every integer r ≥ 3, there are infinitely many integers n with ω(n) = r for which no such representation exists. Further, if a representation holds, we obtain the bounds

2 ≤ x ≤ 9 + 3e^γ log log n,       nx/(φ(n)x − n)≤ y ≤2nx/(φ(n)x − n).

In addition, if x ≥ 3(n−1)(e^γ log log n+3)/n , then rad(n) ≤ (4^2r−4^2r-1)/3 , where γ is Euler’s constant and r = ω(n).