A simple information theoretic argument, based on entropy, suggests that (almost all) numbers of N bits have a multiple of 2xN bits with a N/10 binary hamming weight (HW).
(below, log() is in base 2).
It goes like this: if the probability of a bit of being 1 is "p" (and of being 0 is q=1p), then the information of such a bit is:
p*log(1/p) +q*log(1/q). (Thus the information content of a Nbits number with p=1/2 is N bits). The information content of a number with "loaded" bits (i.e. p != 1/2) is lower perbit, thus the total number of bits must be increased to compensate. It turns out that when p=1/10, the number of bits needs to be roughly doubled (because 1/10*log(10)+9/10*log(10/9)==0.469 ~= 0.5). So overall, a 5x reduction of the HW is achieved through a doubling of the bitlength.
This argument holds for arbitrary numbers Nbits in length. OTOH primorials are not arbitrary numbers at all. In fact, the information content of a primorial of Nbits in length is not N bits but log(N) bits, thus much much lower. (this can be seen like this: if I want to transmit you a primorial Q#, it's enough to transmit just the value Q, which is O(log(Q#)) in size.
Because of this "low information content" of primorials, it is not excluded that they might have muchlower HW multiples then expected from the above informationtheoretic argument. (but not implied either).
Last fiddled with by preda on 20200730 at 02:21
