Problem 5

Let

P(x)

be a polynomial of degree

n>1

with integer coefficients, and let

k

be a positive integer. Define

Q(x)=P(P(\ldots P(x)\ldots))

, where

P

occurs

k

times. Prove that there are at most

n

integers

t

such that

Q(t)=t

Step 5 of 8: Compare two 2-cycles by divisibility

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|\alpha-a|=|\beta-b|,\qquad |\alpha-b|=|\beta-a|

Detailed analysis

Fix one genuine 2-cycle $P(a)=b$ , $P(b)=a$ . For another fixed point $P^2(\alpha)=\alpha$ , write $\beta=P(\alpha)$ , so $P(\beta)=\alpha$ . Applying the divisibility lemma to $(\alpha,a)$ and $(\beta,b)$ in both directions shows that $\alpha-a$ and $\beta-b$ divide each other, and likewise $\alpha-b$ and $\beta-a$ divide each other. Hence the displayed absolute-value equalities hold.