Problems & Puzzles: Conjectures

Conjecture 50 : n = pq - rs

Patrick Capelle sends the following conjectures, closely related in form to the Conjecture 49 :

Conjecture A : Every integer n can be written in infinitely many ways as n = p.q - r.s , where p, q, r, s are primes or 1.

Conjecture B : Every integer is the difference of two semiprimes in infinitely many ways.

Conjecture C : Every integer different from 0 is the difference of two consecutive semiprimes in infinitely many ways.

Questions :

1. Do you have some comments about any of the three conjectures ?

2. Can you prove any of them or find a counterexample ?

Anton Vrba comments:
My comment for Parts A and B
Consider:  Dirichlet's Theorem on Primes in Arithmetic Progressions (1837)  (
If n and s are relatively prime positive integers, then the arithmetic progression:  n, n+s, n+2s, n+3s, ...n+x*s,...  contains infinitely many primes.
Similarely, the above series also contains  infinitely many semi-primes and composite with 3,4 5,... prime factors.
Further, in the above series a fair proportion, that is infinite many cases are of the form n+r*were r is a prime.
Hence n+r*s = p (and n+r*s = p*q for the semi-prime cases)
So, in my opinion, Patrick Capelle conjectures A and B is a reformulation of Dirichlet's Theorem on Primes in Arithmetic Progressions, hence it is true.


Patrick Capelle has continued exploring the concept of semiprimes and conjectures based on them. Perhaps you'll find interesting his last delivery.


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