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Investigation of one class of diophantine equations

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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. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp. 831–836, June, 2000.

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