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 8 of 8: Undo the normalization and verify the converse

Previous step Next step

P(x)=x^n+k\ \text{or}\ P(x)=-x^n+k\quad(n\ge0\text{ integer},\ k\in\mathbb Z)

Detailed analysis

Steps 1–6 show the normalized polynomial is $x^n$ , so undoing the sign and shift of step 1 gives $P(x)=\pm x^n+k$ for some integer $n\ge0$ and integer $k$ . Conversely, if $P(x)=\pm x^n+k$ and $P(s),P(t)$ are integers then $s^n,t^n$ are integers, so $(st)^n=s^nt^n$ is an integer and $P(st)=\pm(st)^n+k$ is an integer. Hence these are exactly the polynomials with the required property.