| Summary: | 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>.
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