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I was playing around with semi-prime numbers and I made two conjectures, which are:

  1. Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime.

  2. There are infinitely many integers $a$, such that $a,(a+1)$ and $(a+2)$ are semi-primes.

I've written a computer program to verify the conjectures for values up to 700,000 (so, there's a high chance that they are both true).

Can anyone give a proof or counter example for any of these problems, or a link to a paper on the subject?

Thanks!

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    $\begingroup$ Conjecture (1) is easy: every four consecutive integers contains a multiple of four. Conjecture (2) has an associated OEIS entry but I'm not aware of any results or research on it. $\endgroup$ – anon Feb 10 '12 at 21:01
  • $\begingroup$ Yes. Thanks! I didn't even think about that ;) Since 5 and 3 are not semi-primes the prof becomes trivial. :) $\endgroup$ – Obinna Okechukwu Feb 10 '12 at 21:06
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There are standard, but hopeless, conjectures in Number Theory that would imply, for example, that there are infinitely many $a$ such that $a=3p$, $a+1=2q$, and $a+2=5r$, where $p,q,r$ are all prime. A search for prime $k$-tuples conjecture should get you started on the literature.

Some idea of the current status can be gleaned from Daniel A. Goldston, Sidney W. Graham, Janos Pintz and Cem Y. Yıldırım, Small Gaps Between Almost Primes, the Parity Problem, and Some Conjectures of Erdős on Consecutive Integers, Int Math Res Not Volume 2011, Issue 7, Pp. 1439-1450, possibly available at http://imrn.oxfordjournals.org/content/2011/7/1439.short.

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  • $\begingroup$ After Green & Tao's "Linear Equations in Primes", I don't think Dickson's conjecture is hopeless anymore. Unsolved, yes, and extremely difficult. But it no longer seems unapproachable. $\endgroup$ – Charles Mar 13 '12 at 15:42

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