Suppose that $k\geq2$ and $0<x<y$ and $y^2-x^2\bigm|2^ky-1$ and $2^k-1\bigm|y-1$. Is it necessarily the case that $x=1$ and $y=2^k$?

I've tested this up to $k\leq50$ and $y\leq10000$ but I haven't been able to make much progress with standard number theoretic techniques.

If $k=1$ then there are infinitely many solutions of the form $x=y-1$.


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