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.  

Note that if $y$ is a perfect square then $y\equiv0,1\pmod4$:


*

*$z\equiv0\pmod4 \implies z^2\equiv 0^2\equiv0\pmod4$

*$z\equiv1\pmod4 \implies z^2\equiv 1^2\equiv1\pmod4$

*$z\equiv2\pmod4 \implies z^2\equiv 2^2\equiv0\pmod4$

*$z\equiv3\pmod4 \implies z^2\equiv 3^2\equiv1\pmod4$


Therefore:
If $n\equiv0\pmod4$ and $c\equiv2,3\pmod4$ then $nx+c\equiv2,3\pmod4$ hence not a perfect square.
