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 2 of 8: Hit a large prime by the intermediate value theorem

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p>\sum_{i=1}^n|a_i|,\quad p\ \text{prime},\qquad \exists\, t\in\mathbb R:\ P(t)=p

Detailed analysis

Fix a prime $p$ larger than $\sum_{i=1}^n|a_i|$ . Since $P(0)=0$ , $P$ is continuous, and $a_n>0$ makes $P(x)\to\infty$ as $x\to\infty$ , the intermediate value theorem gives a real $t>0$ with $P(t)=p$ .