Let $n$ is positive integer number, does always exist a prime number between $n(n+1)$ and $(n+1)(n+2)$?
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13$\begingroup$ The difference between $n(n+1)$ and $(n+1)(n+2)$ is $2(n+1) = 2n + 2$. Also, $n(n+1)$ is $n$ larger than $n^2$, plus $(n+1)(n+2) = n^2 + 3n + 2$ is $n + 1$ larger than $(n+1)^2$. Thus, what you're asking is very similar to Legendre's conjecture and, as such, I suspect it's also currently unknown. $\endgroup$– John OmielanCommented Jun 15, 2021 at 2:07
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4$\begingroup$ It is suspected, but far from being proven. $\endgroup$– GH from MOCommented Jun 15, 2021 at 2:28
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$\begingroup$ It is likely, that, letting $\Delta_{n}:=(n+1)(n+2)-n(n+1)$, one should have $\pi((n+1)(n+2))-\pi(n(n+1))>(1+o(1))\frac{\pi(\Delta_{n})^{2}}{\Delta_{n}}$. $\endgroup$– Sylvain JULIENCommented Jun 15, 2021 at 8:00
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1$\begingroup$ See also the marginally stronger conjecture of Oppermann: en.wikipedia.org/wiki/Oppermann%27s_conjecture $\endgroup$– Terry TaoCommented Oct 3, 2023 at 16:25
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2 Answers
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It is widely believed that there indeed exists a prime between $n(n+1)$ and $(n+1)(n+2)$ for every integer $n\geq 1$. This appears to be beyond the scope of existing methods, however. Even the easier problem of find a prime between $n^3$ and $(n+1)^3$ for all integers $n\geq 1$ appears to be beyond the scope of existing methods. Dudek recently proved that there exists a prime between $n^3$ and $(n+1)^3$ for all $n\geq \exp\exp(33.217)$. Such an explicit threshold is not yet known to exist for primes between $n^2$ and $(n+1)^2$; your proposed problem has a similar predicament.
answered Jun 17, 2021 at 4:23
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11$\begingroup$ We don't know that a threshold exists for the OP's question. That is, currently we cannot exclude the possibility that infinitely many counterexamples $n$ exist. $\endgroup$ Commented Jun 17, 2021 at 7:42
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$\begingroup$ @GHfromMO So the same would hold for Legendre's problem? I didn't realize that so little was known! $\endgroup$ Commented Jun 17, 2021 at 13:13
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2$\begingroup$ Yes, same for Legendre's problem. We don't even know that, for large $x$, there always exists a prime in $[x,x+x^{0.501}]$, which is much weaker than either of the two conjectures. $\endgroup$ Commented Jun 17, 2021 at 15:11
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2$\begingroup$ @GHfromMO Good point...I should've caught that. I'll adjust my answer. $\endgroup$ Commented Jun 17, 2021 at 17:18
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The best result is due to Baker, Harman, and Pintz. They proved the existence of a prime in $[x,x+x^{0.525}]$ for sufficiently large $x$.
