Alain Rochelli wrote on
April 1, 2021, the following
conjecture:
Semiprimes p * q for which p
<= q < p^2 are explained in the sequence A251728.
Let SP(A,B) the
number of semiprimes between A and B and PI(A,B) the number
of primes also between A and B, we conjecture that: The
ratio R(A,B) = SP(A,B) / PI(A,B) is converging to log(2) =
0.693147 (with a sufficiently representative sample [A,B]).
For example, I computed with B-A = 2*10^6 :
A=10^9 ; SP=66222 ; PI=96417 ;
R=0.68683
A=10^12 ; SP=50318 ; PI=72413 ; R=0.69488
A=10^15 ; SP=39874 ; PI=57893 ; R=0.68875
A=10^17 ; SP=35792 ; PI=50952 ; R=0.70247
A=10^18 ; SP=33320 ; PI=48427 ; R=0.68805
Q1. Can you prove this conjecture ?
Q2. Send your own verifications.
On June 14, 2021, Jan van Delden
wrote:
Just to let you know.
Conjecture 90 can be proven. However, I do rely on an article I
found and I do think the Authors made a serious mistake in the
derivation of a density.
Before I give you my contribution, I’d rather check my findings
with these authors. Furthermore I think that a full proof does
take up some typesetting for readability (Which takes some time,
which I don’t have to spare right now).
You can find the article, a
pdf, by just browsing:
On distribution of
semiprimes
(Written by two Russians,
hence the English in the title is a bit ‘off’).
I had to change the bound y^(1/4) to y^(1/3) and correct for the
limited number of primes q such that p<=q<p^2.
To be more specific, the given asymptotic formula in the article
for g*(y) is off, and should be:
g*(y)=ln(3)/ln(y)+O(1/ln^2(y))
Which is why I send an email
to the authors, in order to verify this. (Which they hopefully
answer).
***
On June 7, 2021, Giorgios
Kalogeropoulos wrote:
Q2. I extended the existing table for B-A =
2*10^6 :
A=10^20 ;
SP=30035 ; PI=43404 ; R=0.691987
A=10^25 ;
SP=24102 ; PI=34726 ; R=0.694062
A=10^30 ;
SP=19937 ; PI=29054 ; R=0.686205
A=10^35 ;
SP=17311 ; PI=24960 ; R=0.693550
A=10^40 ;
SP=14997 ; PI=21688 ; R=0.691488
A=10^45 ;
SP=13378 ; PI=19450 ; R=0.687815
A=10^50 ;
SP=12044 ; PI=17475 ; R=0.689213
***
On July 5, 2021, Jan
wote again:
Not a full
proof, but I would say ‘close enough’. The main idea is derived
from the previously mentioned article. I deviated from their
approach slightly, for two reasons, I could simplify their
attack and had to correct a few mistakes, which don’t effect
their primary focus, but it does influence their statement with
regard to the distribution of strong semiprimes in an interval.
I wouldn’t mind
if somebody took the trouble to check whether my calculations
are correct; I wouldn’t want to make a mistake in stating that
other people’s work is incorrect, the authors of the original
article haven’t reacted yet.
See this Jan's
file.
***
On August 15, Jan
wrote again:
Would you be so kind as to update the article I send you in July
to the given attachment? (conjecture 90).
It
had a typo in (6.2) and consequently in (6.4g, 64g*), which
luckily doesn’t influence the result, but I would be happy if
there are no errors…
See
this Jan's new file.
***
