Let $F_n$ be the nth Fibonacci number. The order of appearance $z\left(n\right)$ of a natural number n is defined as the smallest positive integer k such that $F_k\equiv 0(modn)$. In this paper, we shall find all positive solutions of the Diophantine equation $z\left(\phi \right(n\left)\right)=n$, where $\phi$ is the Euler totient function.

中文翻译：

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出现阶与欧拉函数之间的迭代不动点

让 $F_{ñ}$是第n个斐波那契数。出现顺序$ž（ñ）$的自然数Ñ被定义为最小的正整数ķ使得$F_{ķ}\equiv 0（模ñ）$。在本文中，我们将找到Diophantine方程的所有正解$ž（\phi （ñ））=ñ$，在哪里 $\phi$ 是Euler上位函数。