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.

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

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


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    $\begingroup$ No, we don't know this. Chen's theorem gives that there are infinitely many primes $p$ for which $p+2$ has at most two prime factors, but the sieve has a ``parity problem" preventing one from getting numbers with exactly two prime factors. In other words, we can say $n(n+2)$ can have two or three prime factors, but not exactly two prime factors (or exactly three, or four). $\endgroup$ – Lucia Sep 6 '15 at 3:19
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    $\begingroup$ @Lucia, note that Joseph is asking about $n$ and $n+1$, not $n$ and $n+2$ (although I suspect the answer is the same). $\endgroup$ – Gerry Myerson Sep 6 '15 at 4:44
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    $\begingroup$ @GerryMyerson: Thanks, I didn't spot that, but the same comments apply -- we don't know how to make the sequence $n(n+1)$ have exactly an even number of prime factors (like $4$). $\endgroup$ – Lucia Sep 6 '15 at 4:54
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    $\begingroup$ @Lucia: Thank you for that definitive answer, and for pointing to Chen's Theorem. $\endgroup$ – Joseph O'Rourke Sep 6 '15 at 13:41
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    $\begingroup$ @GerhardPaseman: Indeed, as I asked about here: mathoverflow.net/q/212026/6043 $\endgroup$ – Charles Sep 8 '15 at 15:21

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