Finding solution of this equation in set of positive integers.

Question

Could you help me to obtain solutions of the equation $2^{2k+1}-n^2 =1$ in set of positive integers, where $k$ and $n$ are positive integers. In case there is no solution, how to prove it.

Thanks in advance -Richard Sieman

Answer 1

There is no solution:

Let $k \geq 1$. We have $2^{2k+1}-1=1+2+4+\ldots+2^{2k} \equiv 3 \mod 4$ which should be the square of an integer. But the only squares $\mod 4$ are $0$ and $1$.