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
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 semiprimes 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*s were r is a
prime.
Hence n+r*s = p (and
n+r*s = p*q for the semiprime
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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