Investigation of one class of diophantine equations

Abstract

We consider the problem of existence of solutions of the equation$\frac{X}{Y} + \frac{Y}{Z} + \frac{Z}{X} = m$ in natural numbers for differentm∈N. We prove that this equation possesses solutions in natural numbers form=a ²+5,a∈Z, and does not have solutions ifm=4p ²,p∈N, andp is not divisible by 3. We also prove that, forn≥12, the equation

$$\frac{{b_1 }}{{b_2 }} + \frac{{b_2 }}{{b_3 }} + \cdots + \frac{{b_{n - 1} }}{{b_n }} + \frac{{b_n }}{{b_1 }} = m$$

possesses solutions in natural numbers if and only ifm≥n,m∈N.