How can it be proven that integers of the form $n=6jk\pm j \pm k;\ j,k\in \mathbb N^*,$ are the only ones which (when multiplied by $6$) correspond to multiples of $6$ not between twin primes? How do we know that no other integers exist such that when they are multiplied by $6$ their products are also not between twin primes?
A few years ago, I asked this question hoping that someone might have some advice about tackling Diophantine equations of this sort. No such luck.
Specifically, the question has to do with finding integer solutions $n, j, k \in \mathbb N^*$ to the following equation $$ n=6jk\pm j\pm k $$
Since every pair of twin primes must have a multiple of $6$ between them, it is a simple matter to show that integers, $n$, of this form correspond to multiples of $6$ that don't flag a twin prime pair - i.e. $6n+1$ or $6n-1$ is not prime.
The reasoning is simply that whenever a multiple of $6$ is divisible by an integer one less or one greater than some other multiple of $6$, then there is a nearby multiple of $6$ which is adjacent to a multiple of that same number one less or one greater. For instance, $30$ is divisible by $5$, so neither $24$ nor $36$ can fall between twin primes since they are adjacent to $25$ and $35$ respectively. Likewise, $210$ is a multiple of $6$ which is not between twin primes since $209=11\times 19$, and this can be determined with the formula since $210=6\times 35$, $35=33+2$, and $33$ is a multiple of $11=6\times 2 -1$.
In other words, this equation acts as a sieve selecting out multiples of $6$ which definitely do not neighbor twin primes. Therefore, this line of reasoning imposes a necessary condition on any potential twin prime candidates.
Is this condition not only necessary but sufficient?
OEIS contributor Jon Perry seems to think that it is sufficient since he claims here that "6n-1 and 6n+1 are twin primes iff n is not of the form 6ab +- a +- b."
The proof of the conditional is fairly straightforward (as I have already explained), but the converse is far less obvious to me.
How can one prove that "If $n$ is not of the form $6ab\pm a \pm b$, then $6n-1$ and $6n+1$ are twin primes"?