2025/04/28 by Christian Hercher, Hercher, Christian
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2504.19915
openalex publication_date 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
While solving a special case of a question of Erdős and Graham Steinerberger asks for all integers n with ϕ(n)=(2)/(3) ⋅ (n+1). He discovered the solutions n∈\5, 5 ⋅ 7, 5⋅ 7⋅ 37, 5⋅ 7⋅ 37⋅ 1297\ and found that any additional solution must be greater than 1010. 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≥ 1014.