TITLE: For a  polynomial $P$ for which $P(x+5) - P(x) = 2 ,\forall x$. What is the least possible value of $P(4) - P(2)$?
QUESTION [8 upvotes]: There are  infinite number of polynomials $P$ for which $P(x+5) - P(x) = 2,\forall x \in \mathbb{R}$. 
How could we determine the least possible value of $P(4) - P(2)$?

REPLY [2 votes]: Hint: for a polynomial $f(x)$ over a field $K$ of characteristic $ =0,\:$ and $\:0\ne k\in K$
$$ \deg f \le 1\:  \iff\: f(x+k) - f(x) \in K\: \iff\: \dfrac{f(x+k) - f(x)}k\: =\: f'(0)\:=\:\dfrac{f(4)-f(2)}2 $$