The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^na +b$ is the square of an integer. Show that $a = 0$.
I have been trying to show that if $2a+b$ is a perfect square then $4a+b$ or $8a+b$ isn't. Am I going the wrong way? Could you give some hints to proceed with? Do you have any ideas on how to proceed with this question?