October 16, 2025Open Access

On the representation of rational numbers via Euler's totient function

WZWeilin ZhangLanzhou Jiaotong University FCFeng ChenSouth China Agricultural University HLHongjian LiGuangdong University of Foreign Studies

Key Points

Every positive rational number can be represented using Euler's totient function.
The study showcases two specific forms of representation involving positive integers.
Further results and implications of these representations are discussed.
Natural numbers are used in defining the parameters of the representations.

Abstract

Let b>1 be an odd positive integer and k, l N. In this paper, we show that every positive rational number can be written as (m^2) / ( (n^2) ) ^b and (k (m^2-1) ) / (ln^2), where m, n N and is the Euler's totient function. At the end, some further results are discussed.

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