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Two fixed points of a function

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November 9, 2010
functionalgebra unsolvedalgebra

Problem Statement

Suppose that f:{1,2,…,1600}→{1,2,…,1600}f:\{1, 2,\ldots ,1600\}\rightarrow\{1, 2,\ldots ,1600\} satisfies f(1)=1f(1)=1 and f^{2005}(x)=x \text{for}\ x=1,2,\ldots ,1600. (a)(a) Prove that ff has a fixed point different from 11. (b)(b) Find all n>1600n>1600 such that any f:{1,…,n}→{1,…,n}f:\{1,\ldots ,n\}\rightarrow\{1,\ldots ,n\} satisfying the above condition has at least two fixed points.
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