<s>[INST] Are $4ab\pm 1 $ and $(4a^2\pm 1)^2$ coprime? Let $a\ne b$ be two positive integers. Are $4ab+1$ and $(4a^2+1)^2$ coprime always?
Can you find $a$ and $b$ such that they are not coprime?

Edit:
It has been proved that $4ab-1$ is not a divisor of $(4a^2-1)^2$.
Are $4ab-1$ and $(4a^2-1)^2$ always coprime?
 [/INST] When $a=1$ and $b=6$, $4ab+1=25$ and $(4a^2+1)^2=25$.
When $a=1$ and $b=4$, $4ab-1=15$ and $(4a^2-1)^2=9$.
</s>