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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19915 |
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Table of Contents:
- 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 $n\geq 10^{14}$.