A semiprime is a number that is the
product of two (possibly equal) primes.
Define *twin semiprimes* (my terminology) as two consecutive numbers both semiprimes.
For example, $(57,58)$ are twin semiprimes because $57=3 \cdot 19$ and $58=2 \cdot 29$.
And $(622,623)$ are twin semiprimes because $622=2 \cdot 311$ and $623=7 \cdot 89$.
The semiprimes (a.k.a. biprimes or 2-almost primes) are integer sequence A001358.

. Is it known that there are an infinite number of twin semiprimes?Q

At least superficially, the twin semiprimes seem abundant, as this histogram of their frequency out to $n=10^6$ shows:

Chen's Theorem. $\endgroup$ – Joseph O'Rourke Sep 6 '15 at 13:41