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Let <inline-formula><math display="inline"><semantics><msub><mi>F</mi><mi>n</mi></msub></semantics></math></inline-formula> be the <i>n</i>th Fibonacci number. The order 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 natural number <i>n</i> is defined as the smallest positive integer <i>k</i> such that <inline-formula><math display="inline"><semantics><mrow><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></mrow></semantics></math></inline-formula>. In this paper, we shall find all positive solutions of the Diophantine equation <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>(</mo><mi>φ</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mi>φ</mi></semantics></math></inline-formula> is the Euler totient function.