What question did this study set out to answer?

The study aims to explore the decomposition of Euler's totient function values, specifically focusing on cases where ε = φ(n).

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March 10, 2026Open Access

Leikhman's Theorem (on the decomposition of Euler's totient values)

Key Points

The study aims to explore the decomposition of Euler's totient function values, specifically focusing on cases where ε = φ(n).
Examined Euler's totient function φ(n) for numbers ε that can be represented as φ(n)
Identified divisors σ and ω of n such that φ(σ) · φ(ω) = ε
Conducted empirical verification for all ε values up to 10,000,000 through computational experiments
Established that φ(σ) · φ(ω) = ε, with σ and ω being arbitrary divisors of n
Showed that σ and ω do not have to be coprime or multiply to n
Provided trivial cases where σ = 1 and ω = n are valid, contributing to the theorem's applicability

Abstract

AbstractThis paper examines Euler’s totient function φ(n). It is establishedthat for any number ε that can be represented as ε = φ(n), there existdivisors σ, ω of n such that φ(σ) · φ(ω) = ε. Unlike the classicalmultiplicative property, which requires σ · ω = n and gcd(σ, ω) = 1,in this theorem σ and ω are arbitrary divisors of n, not necessarilycoprime and not necessarily multiplying to n; moreover, the equalityholds for all ε in a wide range of values. An empirical verificationis provided for all ε ≤ 10 000 000 using a computational experiment.The trivial cases σ = 1, ω = n and σ = n, ω = 1 are also allowed. Theresult represents a new observation on the structure of Euler’s totientvalues and may serve as a basis for further theoretical research.

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