Investigation of one class of diophantine equations
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...
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| Date: | 2000 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2000
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4478 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865791615210094592 |
|---|---|
| author | Bondarenko, A. V. Бондаренко, А. В. |
| author_facet | Bondarenko, A. V. Бондаренко, А. В. |
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"author": "А. В. Бондаренко",
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| author_sort | Bondarenko, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T20:29:37Z |
| description | 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. |
| first_indexed | 2026-03-24T02:59:42Z |
| format | Article |
| fulltext |
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| id | umjimathkievua-article-4478 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:59:42Z |
| publishDate | 2000 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/26/17ff63ae9425561b7c9e5ed87591c126.pdf |
| spelling | umjimathkievua-article-44782020-03-18T20:29:37Z Investigation of one class of diophantine equations Исследование одного класса диофантовых уравнений Bondarenko, A. V. Бондаренко, А. В. 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. Розглядається питання про існування розв'язків рівняння $\frac{X}{Y} + \frac{Y}{Z} + \frac{Z}{X} = m$ в натуральних числах при різних $m ∈ N$. Доведено, що при $m = a_2 + 5,\; a ∈ Z$, рівняння має розв'язки в натуральних числах, а при $ m = 4p^2,\; p ∈ N$, $р$ не ділиться на 3, не має розв'язків. Також доведено, що при $n ≥ 12$ рівняння $$\frac{{b_1 }}{{b_2 }} + \frac{{b_2 }}{{b_3 }} + \cdots + \frac{{b_{n - 1} }}{{b_n }} + \frac{{b_n }}{{b_1 }} = m$$ має розв'язки в натуральних числах тоді і тільки тоді, коли $m ≥ n, m ∈ N.$ Institute of Mathematics, NAS of Ukraine 2000-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4478 Ukrains’kyi Matematychnyi Zhurnal; Vol. 52 No. 6 (2000); 831–836 Український математичний журнал; Том 52 № 6 (2000); 831–836 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4478/5660 https://umj.imath.kiev.ua/index.php/umj/article/view/4478/5661 Copyright (c) 2000 Bondarenko A. V. |
| spellingShingle | Bondarenko, A. V. Бондаренко, А. В. Investigation of one class of diophantine equations |
| title | Investigation of one class of diophantine equations |
| title_alt | Исследование одного класса диофантовых уравнений |
| title_full | Investigation of one class of diophantine equations |
| title_fullStr | Investigation of one class of diophantine equations |
| title_full_unstemmed | Investigation of one class of diophantine equations |
| title_short | Investigation of one class of diophantine equations |
| title_sort | investigation of one class of diophantine equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4478 |
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