For a  polynomial $P$ for which $P(x+5) - P(x) = 2 ,\forall x$. What is the least possible value of $P(4) - P(2)$? 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 $P(4) - P(2)$?

The condition $P(x+5)-P(x)=2$ for all $x$ can be rewritten as "$(x,y)$ is on the graph of $y=P(x)$ if and only if $(x+5,y+2)$ is."  This implies that if $(x,y)$ is on the graph of $y=P(x)$, so are $(x+5k,y+2k)$ for all $k\in\mathbb{Z}$.  All of those points lie on a line of the form $y=\frac{2}{5}x+b$ for some $b\in\mathbb{R}$, so that line intersects the graph of $y=P(x)$ infinitely many times, or equivalently, $P(x)=\frac{2}{5}x+b$ has infinitely many solutions.
The only way for that to happen with a polynomial $P$ is if $P(x)=\frac{2}{5}x+b$ for all $x\in\mathbb{R}$ (two distinct polynomials can only be equal at finitely many points—if two polynomials are equal at infinitely many points, they must be identically equal).  Thus, $$P(4)-P(2)=\left(\frac{2}{5}\cdot 4+b\right)-\left(\frac{2}{5}\cdot 2+b\right)=\frac{4}{5}.$$
