Find all positive integers n such that $2^2 + 2^5+ 2^n$ is a perfect square.

Question

Find all positive integers n such that $2^2 + 2^5 + 2^n$ is a perfect square. Explain your answer.

Answer 1

We have: $$2^n+36=m^2$$ $$2^n=(m-6)(m+6)$$ Both $m-6$ and $m+6$ must be powers of $2$ (and note that they differ by $12$). Can you continue?

Answer 2

You want to solve $36+2^n=k^2\iff2^n=k^2-6^2\iff2^n=(k-6)(k+6)$ So $\exists a,b \space such\space as\space k-6=2^a, k+6=2^b\space and \space a+b=n$. You make a small table : $2^0=1,2^1=2,2^2=4,2^3=8,2^4=16..$the only values for $2^a$ and $2^b$ with an offset of 12 are $2^a=16$ and $2^b=4$. So the only solution is $n=6$ indeed $16*4=28=2^6$.