Does every linear integer polynomial give a square at some integer? My question is, if you have some function
$$f(x)=nx+c$$
which accepts only integer inputs of $x$, where $n>0$ and $c$ are fixed integer constants, can you always find an $x$ such that
$$f(x)=k^2$$
where $k$ is an integer, i.e. the result is a perfect square.  

No: all squares are equal to $0$ or $1$ modulo $3$, so $n=3$, $c=2$ is a counterexample. Indeed, you can probably find infinitely many such, since the same is true modulo $4$, and probably for larger moduli.
