prove that has infnitely many natural solutions.

Question

Well, the initial problem was to prove that there are many squares that can be expressed in the form but I thought that it would be better to change the problem to

I tried to approach this in many ways like Mathematical induction, but I still have no idea on how to solve this.

Answer 1

The proposition is wrong. $1 + 2^x = y^2 \Longrightarrow 2^x = (y-1)(y+1)$ or we can write $2^x = n(n+2)$ for some $n$. Then both $n$ and $n+2$ has to be $2^{t_i}$, clearly it is wrong when $n>2$.

Answer 2

Since $y^2$ is odd, so is $y$. Thus $y=2k+1$ for some $k$. Therefore we have $$1+2^x=4k^2+4k+1$$ or $$2^x=4k(k+1)$$ from this we need to have both $k$ and $k+1$ powers of $2$ and the only possiblity is $k=1$.