Problems & Puzzles: Conjectures
Conjecture 90. Semiprimes/Primes ->log(2)
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:
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.
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