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The order of appearance (in the Fibonacci sequence) function <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 an arithmetic function defined for a positive integer <i>n</i> 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><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>. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation <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><mi>z</mi><mrow><mo>(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>n</i> is a prime number. In addition, in 2014, Luca and Pomerance proved 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><mo>=</mo><mi>z</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula> has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">lim</mo><munder><mo movablelimits="false" form="prefix">sup</mo><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mrow><mo>(</mo><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>/</mo><msup><mrow><mo>(</mo><mo form="prefix">log</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>, for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>.