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Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positive solutions of the Diophantine equation z(φ(n))=n, where φ is the Euler totient function.

On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function