The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)

Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n<...

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Autores principales: Eva Trojovská, Venkatachalam Kandasamy
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
order of appearance
fibonacci numbers
divisibility
natural density
prime numbers
Mathematics
QA1-939
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spelling oai:doaj.org-article:df08783acfa44ba78d82f642ea9255f82021-11-25T18:17:09ZThe Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)10.3390/math92229122227-7390https://doaj.org/article/df08783acfa44ba78d82f642ea9255f82021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2912https://doaj.org/toc/2227-7390Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula> be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer <i>n</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mrow><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><mi>n</mi><mo>∣</mo><msub><mi>F</mi><mi>k</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Very recently, Trojovská and Venkatachalam proved that, for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> is divisible by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>k</mi></msup></semantics></math></inline-formula>, for almost all integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>k</mi></msup></semantics></math></inline-formula> by any integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, in particular, we prove this conjecture.Eva TrojovskáVenkatachalam KandasamyMDPI AGarticleorder of appearancefibonacci numbersdivisibilitynatural densityprime numbersMathematicsQA1-939ENMathematics, Vol 9, Iss 2912, p 2912 (2021)
institution DOAJ
collection DOAJ
language EN
topic order of appearance
fibonacci numbers
divisibility
natural density
prime numbers
Mathematics
QA1-939
spellingShingle order of appearance
fibonacci numbers
divisibility
natural density
prime numbers
Mathematics
QA1-939
Eva Trojovská
Venkatachalam Kandasamy
The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
description Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula> be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer <i>n</i> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mrow><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><mi>n</mi><mo>∣</mo><msub><mi>F</mi><mi>k</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Very recently, Trojovská and Venkatachalam proved that, for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow></semantics></math></inline-formula> is divisible by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>k</mi></msup></semantics></math></inline-formula>, for almost all integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mi>k</mi></msup></semantics></math></inline-formula> by any integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, in particular, we prove this conjecture.
format article
author Eva Trojovská
Venkatachalam Kandasamy
author_facet Eva Trojovská
Venkatachalam Kandasamy
author_sort Eva Trojovská
title The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
title_short The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
title_full The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
title_fullStr The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
title_full_unstemmed The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of <i>z</i>(<i>n</i>)
title_sort proof of a conjecture on the density of sets related to divisibility properties of <i>z</i>(<i>n</i>)
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/df08783acfa44ba78d82f642ea9255f8
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