Are all those numbers coprime? The values of $4m^2+1$ and $4m^2+4m+5$ for $m\geq{1}$ are (resp.) 5,17,37,... and 13,29,53,... Those numbers seem to be all coprime : how to prove it if it is true, please ? 

Suppose there is a common divisor $d$. Then $d|4m^2+1$ and $d|4m^2+4m+5$ implies that $d|4m^2+4m+5-4m^2-1$, which is $d|4m+4$. Since $4m^2+1$ is an odd number, so is $d$. Therefore $d|m+1$. Also, $d|4(m+1)^2$, so $d|4(m+1)^2-(4m^2+4m+5)$ and then $d|4m-1$.
Now combine $d|4m-1$ and $d|4m+4$. This gives us $d|5$.
