Diophantine equations involving Euler function

2020/01/22 by Hairong Bai, Bai, Hairong
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2001.08246

openalex publication_date 2020/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we show that the equation φ(|x^m-y^m|)=|xⁿ-yⁿ| has no nontrivial solutions in integers x,y,m,n with xy≠0, m>0, n>0 except for the solutions (x,y,m,n)=((2^t-1±1),-(2^t-1∓1),2,1), (-(2^t-1±1),(2^t-1∓1),2,1), where t is a integer with t≥ 2. The equation φ(|\fracx^m-y^mx-y|)=|\fracxⁿ-yⁿx-y| has no nontrivial solutions in integers x,y,m,n with xy≠0, m>0, n>0 except for the solutions (x,y,m,n)=(a±1, -a, 1, 2), (a± i, -a, 2, 1), where a is a integer with i=1,2.

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