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I asked this question: Gaps between numbers of the form $pq$ , and received a very satisfactory answer. Now I'm curious about a related one. We know that we have have arbitrarily long stretches of numbers with no primes, so there is no upper bound on the gap between $p_i$ and $p_{i+1}$, where $p_k$ is the $k$th prime.

For numbers of the form $pq$, i.e., products of two distinct primes, are there also arbitrarily long gaps between them? Or, is there a number $K$, such that for any natural number $n$, at least one of $n, n+1, \ldots, n+K-1$ is a product of two primes?

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For example, consider $K+1$ distinct primes $p_i$. By the Chinese Remainder Theorem there is $N$ such that $N \equiv i \mod p_i^3$ for all $i$ from $1$ to $K+1$. Thus $N - i$ is divisible by $p_i^3$, and in particular is not a product of two primes, for all of these $i$.

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    $\begingroup$ Ooh, that's good. CRT to the rescue, once again. Thank you! $\endgroup$ – G Tony Jacobs May 14 '17 at 19:39

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