Erdős–Graham Problem #411 – A Construction of Six Solution Families for the r=2 Case

Abstract

We construct and verify six infinite families of solutions to the r=2 case of the Erdős–Graham iteration problem (Problem #411). Steinerberger [1] reduced the r=2 case to the equation φ(n) + φ(n+φ(n)) = n and proved that any solution must fall into one of two branches: either the odd part of n lies in S = {1,3,5,7,35,47} (the first branch), or n takes the form 2^ℓ · (8m+7) or 2^ℓ · (6m+5) with specific conditions (the second branch). We verify that every number in the first branch, with starting exponents a_1 = 2 and a_m = 1 for the remaining cases, satisfies the r=2 property. The second branch is not addressed here and remains an open problem. This paper presents a partial result — a complete construction and verification of the first branch — not a complete proof of the full r=2 case.