This is the $y=2^k$ case of this question.
Suppose that $k\geq1$ and $0
Equivalently: Suppose that there are two positive divisors of $2^{2k}-1$ which average to $2^k$. Is it necessarily the case that these two divisors are $2^k-1$ and $2^k+1$?
This is the $y=2^k$ case of this question.
Suppose that $k\geq1$ and $0
Equivalently: Suppose that there are two positive divisors of $2^{2k}-1$ which average to $2^k$. Is it necessarily the case that these two divisors are $2^k-1$ and $2^k+1$?