<s>[INST] For all $n>1$ there are positive $a+b=n$ such that $a+ab+b\in\mathbb P$ For all integers $n>1$ there are positive integers $a,b$ such that $a+b=n$ and such that $a+ab+b\in\mathbb P$.
Tested for all $n\leq 1,000,000$. Hopefully, someone can explore and explain the heuristics about this conjecture.

This question is related to:
Any odd number is of form $a+b$ where $a^2+b^2$ is prime
Does every power of two arise as the difference of two primes?
Most even numbers is a sum $a+b+c+d$ where $a^2+b^2+c^2=d^2$
Natural numbers large enough can be written as $ab+ac+bc$ for some $a,b,c>0$
$\{a+b|a,b\in\mathbb N^+\wedge ma^2+nb^2\in\mathbb P\}=\{k>1|\gcd(k,m+n)=1\}$
Even numbers has the form $a+b$ where $\frac{a^2+b^2}{2}$ is prime
Is every positive integer greater than $2$ the sum of a prime and two squares? 
It's about a relation $R\subseteq \mathbb N^m$, a function 
$f:\mathbb N^m\to \mathbb N$, and an image of a restriction
$\operatorname{Im}(f|R)$.
In Goldbachs conjecture the relation is $p,q\in\mathbb P$, the function is $(p,q)\mapsto p+q$ and the image of the restriction is 
$2\mathbb N\setminus\{2\}$.
Maybe some of the conjectures can be generalized?
 [/INST] This is only a partial answer. Reformulating according to the comment of Crostul, we want to find $AB-1 \in \mathbb P$. Other than the even prime $2$, all primes are odd, so at least one of $A,B$ must be even. For $A+B=N,\ N\ge 6$, any prime generated will have the form $6m\pm 1$. So it is necessary (but not sufficient) that for every $N$, there are some $A,B$ such that $AB-1=6m\pm 1$ for the conjecture to be true.
$AB-1=6m\pm 1\Rightarrow AB\equiv (0,2) \mod{6}$. Any $N\ge 6$ can be split into two addends with that property. The following table lists values of residues $\mod{6}$ for $N,A,B$ that satisfy $N=A+B\mod{6}$ and $AB\equiv (0,2) \mod{6}$ (up to the order of $A,B$).
$$\begin{array}{ccc}
\ N&A&B \\
0&0&0 \\
&2&4 \\ \\
1&0&1 \\
&3&4 \\ \\
2&0&2 \\
3&0&3 \\
&1&2 \\
&4&5 \\ \\
4&0&4 \\
5&0&5 \\
&2&3 \\
\end{array}$$
This shows that any $N$ can be split into addends $A,B$ such that $AB\equiv (0,2) \mod{6}$. In every case, it is possible to obtain addends such that $AB\equiv 0 \mod{6}$. Interestingly, only for $N\equiv (0,3)\mod6 \Rightarrow N\equiv 0 \mod3$ is it possible to obtain addends such that $AB\equiv 2 \mod{6}$. This means that $AB-1$ can generate numbers of the form $6m-1$ from any $N$, but can generate numbers of the form $6m+1$ only if $N\equiv 0\mod3$. It remains open at this point whether the numbers of the form $6m\pm 1$ obtained from a particular $N$ will necessarily feature a prime.
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