I previously asked for feedback on the following conjectures (similar in appearance to Goldbach ‘s and Lemoine’s see: Two new conjectures related to Lemoine's and Goldbach's) : $O = 2n + 1 = p - 2q$ always has a solution in distinct odd primes $p$ and $q$ for $O \ge 5$ $O = 2n + 1 = 2q - p$ always has a solution in $p$ and $q$ with $q < p$, $p$ being an odd prime and $q$ being 1 or an odd prime and $O \ge 1$
Those two conjectures, even though they do not seem easy to prove, are very weak: the number of possible solutions is possibly infinite. (Since increases in $p$ and $q$ have opposite effect.) Also, while it is possible by concrete calculations to increase the domain of proven validity, it is not possible to find a concrete counter example since a solution may lay with higher values of $p$ and $q$.
Since they are so easy to satisfy, I thought that they may become more interesting if they were generalized by changing the component $2q$ into $2nq$. It is easy to see that in $O = p-2nq$ we will not have solutions in primes $p$ and $q$ as long as $n$ and $O$ have common divisors. So the revised conjectures become:
For $O$, odd natural number, and $n$ ,natural number ,without common divisors: $O = p – 2nq$ always has a solution in distinct odd primes $p$ and $q$ for $o \ge 5$ $O = 2nq – p$ always has a solution in $p$ and $q$ with $q < p$, $p$ being an odd prime and $q$ being 1 or an odd prime and $o \ge 1$
I verified that it is the case for all $O < 2000000$ and $n < 1000$
But better, a further generalization is possible that envelops these conjectures:
$A = Bp – Cq$ where the natural numbers $A,B,C$ have no common divisor and one of them is even has solutions in $p$ and $q$ odd primes or 1
I verified this with values of $A, B$ and $C$ ranging from 1 to 1000.