Investigation of one class of diophantine equations

  • A. V. Bondarenko

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 differentmN. We prove that this equation possesses solutions in natural numbers form=a 2+5,aZ, and does not have solutions ifm=4p 2,pN, 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 ifmn,mN.
Published
25.06.2000
How to Cite
BondarenkoA. V. “Investigation of One Class of Diophantine Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 52, no. 6, June 2000, pp. 831–836, https://umj.imath.kiev.ua/index.php/umj/article/view/4478.
Section
Research articles