Problem 5

Find all polynomials

P(x)

with integer coefficients such that for all real numbers

s

and

t

, if

P(s)

and

P(t)

are both integers, then

P(st)

is also an integer.

Step 3 of 8: Form a common-root gcd

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P(2)\in\mathbb Z,\ P(t)=p\in\mathbb Z\ \Longrightarrow\ P(2t)\in\mathbb Z,\qquad f(x)=\gcd\big(P(x)-p,\ P(2x)-P(2t)\big)

Detailed analysis

Since $P$ has integer coefficients, $P(2)$ is an integer; together with $P(t)=p\in\mathbb Z$ , the hypothesis applied to $s=2$ gives $P(2t)\in\mathbb Z$ . Both $P(x)-p$ and $P(2x)-P(2t)$ are polynomials with integer coefficients vanishing at $x=t$ , so their gcd $f(x)$ (computed over $\mathbb Q$ ) is a non-constant polynomial with rational coefficients dividing both.