MathDB

Let

A=\{1,2,3,\cdots ,17\}

. A mapping

f:A\rightarrow A

is defined as follows:

f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))

for

k\in\mathbb{N}

. Suppose that

f

is bijective and that there exists a natural number

M

such that: i) when

m<M

and

1\le i\le 16

, we have

f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}

and

f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}

; ii) when

1\le i\le 16

, we have

f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}

and

f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}

. Find the maximal value of

M

Problem Statement