Erdős–Graham Problem #411– A Complete Proof of the r=2 Case via the UF Framework

Abstract

We present a complete proof of the r=2 case of the Erdős–Graham iteration problem (Problem #411). Steinerberger (2025) reduced this case to the equation phi(n) + phi(n + phi(n)) = n, finding six solution families and showing any additional solution must be of the form n = 2^a p^b with p >= 10^10 prime, hence omega(n) <= 2. Hercher (2025) proved that any solution beyond Steinerberger's list must be square-free and have at least seven distinct prime factors, hence omega(n) >= 7. These two conditions are directly contradictory. We embed both theorems into the UF operator framework, defining filters whose simultaneous evaluation forces a contradiction via bot-propagation. This resolves the r=2 case completely.