# Is there any other special numbers?

$34$ 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.

• – CY Aries May 4 '17 at 14:31
• Note: it is common to say that $n$ is semiprime if it is the product of $2$ (albeit not necessarily distinct) prime numbers. – Omnomnomnom May 4 '17 at 14:31
• @CYKwong your link doesn't work. It seems that you're saying $86$ is "special", though. – Omnomnomnom May 4 '17 at 14:32
• The link is sosmath.com/tables/factor/factor.html, and there are more: e.q. $93,94,95$ – gammatester May 4 '17 at 14:35
• 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 $10^6$. – Michael Seifert May 4 '17 at 14:59

## 1 Answer

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.