$34$ 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.

  • $\begingroup$ sosmath.com/tables/factor/factor.htm 85,86,87 $\endgroup$ – CY Aries May 4 '17 at 14:31
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.