三元变系数欧拉函数方程φ(abc)=φ(a)+5φ(b)+7φ(c)的正整数解

Positive Integer Solution of Ternary Variable Coefficient Euler Function Equation φ(abc)=φ(a)+5φ(b)+7φ(c)

摘要: 对于任意正整数解n，欧拉函数φ(n)表示从1,2,...,n-1中与n互素的正整数解的个数。欧拉函数方程是数论及其应用中的一个重要的研究论题，不定方程的一元变系数和多元变系数问题使得我们延展出很多新的研究方法，对有关包含欧拉函数方程可解性的研究有着重要的理论意义和应用价值。利用初等数论相关内容及计算方法，研究了系数为奇数的三元欧拉函数方程φ(abc)=φ(a)+5φ(b)+7φ(c)的可解性，其中φ(n)为欧拉函数。得到了该方程共计25组正整数解。

Abstract: For any positive integer solution n, Euler’s function φ(n) represents the number of positive integer solutions that are prime to n from 1,2,...,n-1. Euler’s function equation is an important research topic in number theory and its applications. The univariate variable coefficients and multivariate variable coefficients of indeterminate equations have allowed us to extend many new research methods and research on the solvability of equations containing Euler functions. It has important theoretical significance and application value. Using the related content and calculation method of elementary number theory, the solvability of the ternary Euler function equation φ(abc)=φ(a)+5φ(b)+7φ(c) with odd coefficients is studied, where φ(n) is the Euler function. A total of 25 positive integer solutions of the equation are obtained.