A side discussion over on this question has left me curious: is there any $B$ for which it's known that there are infinitely many primes adjacent to $B$-smooth numbers (i.e., for which there are infinitely many primes of the form $n\pm 1$, where all the prime factors of $n$ are $\leq B$)? Of course, the Lenstra-Pomerance-Wagstaff conjecture about the distribution of Mersenne primes (and specifically just the conjecture that there are infinitely many) implies this conjecture in the sharpest possible form (with $B=2$), but I'm wondering if results of this form have been proven for any value of $B$.
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I'm pretty sure the answer is no, on the grounds that the $B$-smooth numbers are an exponentially thin set, and proofs of infinitely many primes are too much to expect in such circumstances. But I yield to anyone who can supply an actual reference to work on the topic.