3
$\begingroup$

Legendre's conjecture states that for every integer $n\ge1$, there is a prime between $n^2$ and $(n+1)^2$. This is an important problem about prime numbers that has not been solved for a long time, but I wonder if Legendre's conjecture can be proven for semiprime numbers rather than prime numbers?

Thanks

Improve this question
$\endgroup$

1 Answer 1

Reset to default
7
$\begingroup$

See Chen's 1975 article "On the Distribution of Almost Primes in an Interval." This proves Legendre's conjecture for almost primes (primes or semiprimes) and for sufficiently large $n$. See also a tighter bound in Wu's "Almost primes in short intervals" from 2010 which shows that for sufficiently large $x$, there is always an almost prime in $(x − x^{101/232}, x]$.

Improve this answer
$\endgroup$
4
  • $\begingroup$ I couldn't find this article on the internet, could you please send me the link? $\endgroup$
    – Egehan Eren
    Commented Nov 13, 2023 at 11:57
  • 1
    $\begingroup$ @EgehanEren sciengine.com/Math%20A0/doi/10.1360/ya1975-18-5-611 seems to have a copy. $\endgroup$
    – JoshuaZ
    Commented Nov 13, 2023 at 12:04
  • 1
    $\begingroup$ I wasn't able to check Chen's article, but Wu's only talks about almost primes, that is numbers which are either prime or semiprime. That alone doesn't imply that those intervals contain semiprimes. Does Chen actually have a result for semiprimes? $\endgroup$
    – Wojowu
    Commented Nov 13, 2023 at 17:12
  • 1
    $\begingroup$ @Wojowu Ah, yes you are correct. They both only prove their results for almost primes, not semiprimes themselves. I don't know how easy it would be to modify to just semiprimes. I will edit my answer to reflect that. $\endgroup$
    – JoshuaZ
    Commented Nov 13, 2023 at 17:34

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .