Given any even perfect number $p$ we may notice that the difference between $p$ and closest power of two is also a power of two.
Some examples:
$8-6 = 6-4 = 2$
$32-28 = 4 = 2^2$
$512-496 = 16 = 2^4$
$8192-8128 = 64 = 2^{6}$
$...$
$33554432-33550336 = 4096 = 2^{12}$
$8589934592 - 8589869056 = 65536 = 2^{16}$
...and so on.
So, the questions are:
- Was this fact known before?
- Could we find an even perfect number which breaks the sequence?
- Is this sequence infinite?