On the representation of rational numbers via Euler's totient function
Weilin Zhang, Fengyuan Chen, Hongjian Li, Pingzhi Yuan
TL;DR
The paper proves two universal totient-based representations for all positive rationals: $q=\\dfrac{\\varphi(m^{2})}{(\\varphi(n^{2}))^{b}}$ with odd $b>1$, and $q=\\dfrac{\\varphi(k(m^{2}-1))}{\\varphi(ln^{2})}$. The approach is constructive, employing Dirichlet's theorem on primes in arithmetic progressions to select primes in a way that controls valuations and assembles $m,n$ to realize any $q$, with Pell's equation used in the second form when gcd constraints require it. These results extend prior work by Krachun–Sun, Li–Yuan–Bai, and Vu, and open new directions for classifying quadruples $(a,b,r,s)$ and exploring shifted-totient representations. They also pose open problems about universality for other exponent patterns and broader generalizations, inviting further research in totient-based rational representations.
Abstract
Let $b>1$ be an odd positive integer and $k, l \in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $\varphi(m^{2})/(\varphi(n^{2}))^{b}$ and $\varphi(k(m^{2}-1))/\varphi(ln^{2})$, where $m, n\in \mathbb{N}$ and $\varphi$ is the Euler's totient function. At the end, some further results are discussed.