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 ?

My comment for Parts A and B

Consider:  Dirichlet's Theorem on Primes in Arithmetic Progressions (1837)  (http://primes.utm.edu/notes/Dirichlet.html)

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.

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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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