Given an integer of the form: $N=(2^p+k)$ where
- $N$ is a square number.
- $k \leq p$
- $k$ and $p$ are positive integers.
How difficult is it to prove that there are infinite numbers $N$ that satisfy the above constraint? Is something like this already known?
Best Answer
Following on from quarague's answer, the spacing of squares at that point is $2\cdot \sqrt 2 \cdot 2^q+1$ and we need to be within $2q$ of one, so the chance of a particular $q$ satisfying it is $\frac {2q}{2^{q+1}\sqrt 2+1}$. If we sum this to infinity we get $\sqrt 2$ so we only expect a few.
$$p=3, 3^2=2^3+1\\ p=5, 6^2=2^5+4$$ may be all there are. There are no others by $p=95$, where Excel runs out of precision.