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:

 

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.

***

 

 

 

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