# Cycling Through The Polynomial Shuffle

Number Theory Level 5

Suppose $$f(x)$$ is a polynomial with integer coefficients which is bijective modulo $$49$$, which means that $$x\not \equiv y \pmod {49} \Leftrightarrow f(x)\not \equiv f(y) \pmod {49}.$$ We define the composition powers $$f^{(i)}(x)$$ by setting $$f^{(1)}=f$$ and $$f^{(i+1)}(x)=f(f^{(i)}(x)).$$ For a given $$f,$$ its order (in modulo 49) is defined as the smallest positive integer $$n,$$ such that $$f^{(n)}(x)\equiv x \pmod {49}$$ for all integers $$x.$$

What is the largest possible order of $$f?$$

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