Are there infinitely many primes next to smooth numbers?

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.

Surface integral over ellipsoid

No Smooth Onto Map from Circle to Torus

Comparing powers without logarithms

Is the trace of the product of two positive semidefinite matrices always nonnegative?

Eigenvalues of tridiagonal symmetric matrix with diagonal entries 2 and subdiagonal entries 1

What does it mean to have a lone plus sign in the exponent/superscript (Modified Weiszfeld algorithm)

Is it possible to prove that $3$ is a primitive root of any Fermat prime without quadratic reciprocity?

Define an infinite subset of primes such that the sum of reciprocals converges

What's an example of a vector space that doesn't have a basis if we don't accept Choice?

Prove: If $N\lhd G$ and $N$ and $\frac{G}{N'}$ are nilpotent, then $G$ is nilpotent.

Complexity of counting the number of triangles of a graph

Does there exist a polynomial $f(x)$ with real coefficients such that $f(x)^2$ has fewer nonzero coefficients than $f(x)$?