Find all rational real numbers $x$ for which $\log_2(x^2 - 4x - 1)$ is a whole number.

Question

Find all rational real numbers $x$ for which $\log_2(x^2 - 4x - 1)$ is a whole number.

The results are supposed to be: $x_1 = 5$, $x_2 = -1$, $x_3 = 17/4$, $x_4= -1/4$.

Answer 1

Put $y = x- 2 \implies x^2-4x-1 = 2^m \implies (x-2)^2 = 2^m+5\implies y^2 = 2^m+5$. This shows $y$ must be an integer since $2^m+5$ is an integer. If $m$ is odd, say $m = 2k+1\implies (y-1)(y+1) = 2(4^k+2)= 4(2^{2k-1}+1)$. Thus $y$ is odd. So put $y = 2r+1\implies 4r(r+1) = 4(2^{2k-1}+1)\implies r(r+1) = 2^{2k-1}+1$. This equation has no integer solution since the left side is even while the right side is odd. Thus $m$ must be even, then put $m = 2s\implies (y-2^s)(y+2^s) = 5$. I am sure you can take it from here...