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: | , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/df08783acfa44ba78d82f642ea9255f8 |
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| Sumario: | 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. |
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