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$34$ is quite an interesting number. It's the product of 2 different prime numbers: $2$ and $17$. Also, $34-1$ and $34+1$ also the products of 2 different prime numbers, which are $(3)$$(11)$ and $(5)$$(7)$ respectively.

We have a definition:

Definition: A positive integer n is called special if each of n, n-1 and n+1 is the product of 2 distinct prime numbers.

Question: Is there any special number beside $34$?

Note: I have figured out that if a number is special, then it must be even, but I don't khow what to do next.

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  • $\begingroup$ sosmath.com/tables/factor/factor.htm 85,86,87 $\endgroup$ – CY Aries May 4 '17 at 14:31
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    $\begingroup$ Note: it is common to say that $n$ is semiprime if it is the product of $2$ (albeit not necessarily distinct) prime numbers. $\endgroup$ – Omnomnomnom May 4 '17 at 14:31
  • $\begingroup$ @CYKwong your link doesn't work. It seems that you're saying $86$ is "special", though. $\endgroup$ – Omnomnomnom May 4 '17 at 14:32
  • $\begingroup$ The link is sosmath.com/tables/factor/factor.html, and there are more: e.q. $93,94,95$ $\endgroup$ – gammatester May 4 '17 at 14:35
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    $\begingroup$ A quick computer search finds 13 special numbers less than $10^3$, 71 less than $10^4$, 379 less than $10^5$, and 2377 less than $10^6$. $\endgroup$ – Michael Seifert May 4 '17 at 14:59
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The conjecture is, that there are infinitely many positive integers $n$ such that all three numbers $(n-1,n,n+1)$ are product of two different primes. Thus such numbers might not be so special after all. A slightly more general notion is the notion of a semiprime, which is a natural number that is the product of two (not necessarily distinct) prime numbers. For the corresponding conjecture see this question.

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