On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
Let <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula> be the <i>n</i>th Fibonacci number. The order of appearance <inline-formula>...
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MDPI AG
2020-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/10/1796 |
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| author | Štěpán Hubálovský Eva Trojovská |
| author_facet | Štěpán Hubálovský Eva Trojovská |
| author_sort | Štěpán Hubálovský |
| collection | DOAJ |
| description | Let <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula> be the <i>n</i>th Fibonacci number. The order of appearance <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a natural number <i>n</i> is defined as the smallest positive integer <i>k</i> such that <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mi>k</mi></msub><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mrow><mo>(</mo><mo form="prefix">mod</mo><mspace width="0.277778em"></mspace><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we shall find all positive solutions of the Diophantine equation <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula> is the Euler totient function. |
| first_indexed | 2024-03-10T15:35:25Z |
| format | Article |
| id | doaj.art-48c512b8d2ef48eaa9b687f2329a0de5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T15:35:25Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-48c512b8d2ef48eaa9b687f2329a0de52023-11-20T17:18:51ZengMDPI AGMathematics2227-73902020-10-01810179610.3390/math8101796On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient FunctionŠtěpán Hubálovský0Eva Trojovská1Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicDepartment of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech RepublicLet <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula> be the <i>n</i>th Fibonacci number. The order of appearance <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> of a natural number <i>n</i> is defined as the smallest positive integer <i>k</i> such that <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mi>k</mi></msub><mo>≡</mo><mn>0</mn><mspace width="4.44443pt"></mspace><mrow><mo>(</mo><mo form="prefix">mod</mo><mspace width="0.277778em"></mspace><mi>n</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In this paper, we shall find all positive solutions of the Diophantine equation <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula> is the Euler totient function.https://www.mdpi.com/2227-7390/8/10/1796Fibonacci numbersorder of appearanceEuler totient functionfixed pointsDiophantine equations |
| spellingShingle | Štěpán Hubálovský Eva Trojovská On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function Mathematics Fibonacci numbers order of appearance Euler totient function fixed points Diophantine equations |
| title | On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function |
| title_full | On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function |
| title_fullStr | On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function |
| title_full_unstemmed | On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function |
| title_short | On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function |
| title_sort | on fixed points of iterations between the order of appearance and the euler totient function |
| topic | Fibonacci numbers order of appearance Euler totient function fixed points Diophantine equations |
| url | https://www.mdpi.com/2227-7390/8/10/1796 |
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