On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence
Let <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>≥...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/1/124 |
| _version_ | 1818846353676042240 |
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| author | Eva Trojovská |
| author_facet | Eva Trojovská |
| author_sort | Eva Trojovská |
| collection | DOAJ |
| description | Let <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </semantics> </math> </inline-formula> be the sequence of the Fibonacci numbers. The order (or rank) 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 positive integer <i>n</i> is defined as the smallest positive integer <i>m</i> such that <i>n</i> divides <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>. In 1975, Sallé proved that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>, for all positive integers <i>n</i>. In this paper, we shall solve the Diophantine equation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> for positive integers <i>n</i> and <i>k</i>. |
| first_indexed | 2024-12-19T05:44:12Z |
| format | Article |
| id | doaj.art-f054f8be98b34918adc990cdb32215e2 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-19T05:44:12Z |
| publishDate | 2020-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-f054f8be98b34918adc990cdb32215e22022-12-21T20:33:54ZengMDPI AGMathematics2227-73902020-01-018112410.3390/math8010124math8010124On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci SequenceEva Trojovská0Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech RepublicLet <inline-formula> <math display="inline"> <semantics> <msub> <mrow> <mo>(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </semantics> </math> </inline-formula> be the sequence of the Fibonacci numbers. The order (or rank) 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 positive integer <i>n</i> is defined as the smallest positive integer <i>m</i> such that <i>n</i> divides <inline-formula> <math display="inline"> <semantics> <msub> <mi>F</mi> <mi>m</mi> </msub> </semantics> </math> </inline-formula>. In 1975, Sallé proved that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>≤</mo> <mn>2</mn> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>, for all positive integers <i>n</i>. In this paper, we shall solve the Diophantine equation <inline-formula> <math display="inline"> <semantics> <mrow> <mi>z</mi> <mo>(</mo> <mi>n</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mo>/</mo> <mi>k</mi> <mo>)</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula> for positive integers <i>n</i> and <i>k</i>.https://www.mdpi.com/2227-7390/8/1/124diophantine equationasymptoticfibonacci numbersorder (rank) of appearancep-adic valuation |
| spellingShingle | Eva Trojovská On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence Mathematics diophantine equation asymptotic fibonacci numbers order (rank) of appearance p-adic valuation |
| title | On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence |
| title_full | On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence |
| title_fullStr | On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence |
| title_full_unstemmed | On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence |
| title_short | On the Diophantine Equation <i>z</i>(<i>n</i>) = (2 − 1/<i>k</i>)<i>n</i> Involving the Order of Appearance in the Fibonacci Sequence |
| title_sort | on the diophantine equation i z i i n i 2 1 i k i i n i involving the order of appearance in the fibonacci sequence |
| topic | diophantine equation asymptotic fibonacci numbers order (rank) of appearance p-adic valuation |
| url | https://www.mdpi.com/2227-7390/8/1/124 |
| work_keys_str_mv | AT evatrojovska onthediophantineequationiziini21ikiiniinvolvingtheorderofappearanceinthefibonaccisequence |