13 Sep
2018
13 Sep
'18
9:08 p.m.
On Sep 13, 2018, at 6:58 PM, Keith F. Lynch <kfl@KeithLynch.net> wrote:
Every number in the form 2^k-1, whether prime or not, consists of k adjacent one-bits.
I agree.
Whenever you multiply such a number by anything, you are adding shifted versions of that string of k adjacent one-bits. Due to carries, that results in a sequence of one-bits over all of the overlap region except the rightmost bit of the overlap. You also get one one-bit for each bit to the right of the overlap. And due to carries, you get at least one one-bit to the left of the overlap. So that totals at least k one-bits.
This isn’t how my proof goes — it’s not obvious to me that the carries can’t create some cancellations and end up with fewer 1s. - Cris