On the Parity of the Order of Appearance in the Fibonacci Sequence

• Main Author: Pavel Trojovský
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: order of appearance; Fibonacci numbers; parity; natural density; prime numbers
• Online Access: https://www.mdpi.com/2227-7390/9/16/1928

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><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mo>:</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> 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>=</mo><mo movablelimits="true" form="prefix">min</mo><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><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><mo>}</mo></mrow></semantics></math></inline-formula>. In this paper, among other things, we prove that <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 an even number for almost all positive integers <i>n</i> (i.e., the set of such <i>n</i> has natural density equal to 1).