On Positive Integers $n$ with $φ(n)=\frac{2}{3} \cdot (n+1)$

TL;DR

This work analyzes the equation $\phi(n)=\frac{2}{3}\,(n+1)$, motivated by Erdős–Graham and Steinerberger, and proves structural restrictions on the solutions. The authors show every solution is square-free and that distinct prime factors $p,r$ of $n$ satisfy $p\nmid (r-1)$, enabling a finite classification when the number of prime factors is small. They exhaustively solve the cases with up to four prime factors, obtaining exactly $n\in\{5,35,1295,1679615\}$, and prove that for any fixed number of primes there are only finitely many solutions. A computer-assisted search extends these bounds, ruling out five- and six-prime solutions below $10^{14}$ and establishing that any new solution must have at least seven prime factors, with a practical search limited by computational constraints. The combined theoretical and computational approach thus strongly supports Steinerberger's conjecture that no further solutions exist beyond the known four, under the explored bounds.

Abstract

While solving a special case of a question of Erdős and Graham Steinerberger asks for all integers $n$ with $φ(n)=\frac{2}{3} \cdot (n+1)$. He discovered the solutions $n\in\{5, 5 \cdot 7, 5\cdot 7\cdot 37, 5\cdot 7\cdot 37\cdot 1297\}$ and found that any additional solution must be greater than $10^{10}$. He conjectured that there are no such additional solutions to this problem. We analyze this problem and prove: *) Every solution $n$ must be square-free. *) If $p$ and $q$ are prime factors of a solution $n$ then $p\nmid (q-1)$. *) Any solution additional to the set given by Steinerberger has to have at least 7 prime factors. *) For any additional solution it holds $n\geq 10^{14}$.

Theorems & Definitions (25)

• Theorem 1
• proof
• Corollary 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof