This paper shows how every positive rational number can be expressed using Euler's totient function, indicating new mathematical relationships.
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 φ(m²)/(φ(n²))ᵇ and φ(k(m²-1))/φ(ln²), where m, n∈ N and φ is the Euler's totient function. At the end, some further results are discussed.
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Zhang et al. (2025) studied this question.
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