<s>[INST] Is the $n$-th prime $p_n$ expressible as the difference of coprime $A, B$ such that the set of prime divisors of $AB$ is $\{p_1, \dots, p_{n-1}\}$? We define recursively 
$p_1=2,p_2=3$
and 
$$p_{n}= \min_{(A,B)\in F_{n-1}}|A-B| $$
Where
$$
\begin{split}
F_n=\{(A,B) |&\gcd (A,B)=1,\quad |A-B| \not =1,
\\\
&\text{both $A$ and $B$ are products of powers of $p_i$ for $i\le n$},
\\\
&\text{for each $i\le n$, either $p_i |A$ or $p_i |B$}\}
\end{split}
$$
Is always $p_n$ the $n-th$ prime?
 [/INST] Barry Cipra has already computed the first few values.
The next couple of numbers $p_n$ are
$13 = 5 \cdot 11 - 2 \cdot 3 \cdot 7$,
$17 = 2 \cdot 7 \cdot 13 - 3 \cdot 5 \cdot 11$,
$19 = 2^2 \cdot 3 \cdot 5 \cdot 17 - 7 \cdot 11 \cdot 13 $,
$23 = 7 \cdot 11 \cdot 17^2 - 2 \cdot 3^2 \cdot 5 \cdot 13 \cdot 19$,
$29 = 3 \cdot 11 \cdot 13^2 \cdot 19 \cdot 23 - 2^{12} \cdot 5 \cdot 7 \cdot 17$.
I don't find such expression for 31 in numbers $\leq 10^{12}$.
The smallest I find is $47 = 3 \cdot 7 \cdot 19^2 \cdot 23 \cdot 29 -
2^5 \cdot 5 \cdot 11 \cdot 13^2 \cdot 17$.
If the abc conjecture is true, there are at most finitely many ways to express the $n$-th
prime as a difference of coprime numbers which are divisible only by the first $n-1$ 
primes.
Addendum: For a more extensive table, see http://www.fermatquotient.com/DiverseMinimas/S=M-N_min (found by Google'ing for the numbers obtained above).
The data supports the assumption that the OP's assertion is unlikely to be true.
</s>