Legendre's conjectureLegendresConjecture2007-01-29 18:07:572007-01-29 18:07:57Conjecture% this is the default PlanetMath preamble. as your knowledge
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(Adrien-Marie Legendre) There is always a prime number between a square number and the next. To put it algebraically, given an integer $n > 0$, there is always a prime $p$ such that $n^2 < p < (n + 1)^2$. Put yet another way, $(\pi((n + 1)^2) - \pi(n^2)) > 0$, where $\pi(x)$ is the prime counting function.
This conjecture was considered unprovable when it was listed in Landau's problems in 1912. Almost a hundred years later, the conjecture remains unproven even as similar conjectures (such as Bertrand's postulate) have been proven.
But progress has been made. Chen Jingrun proved a slightly weaker version of the conjecture: there is either a prime $n^2 < p < (n + 1)^2$ or a semiprime $n^2 < pq < (n + 1)^2$ (where $q$ is a prime unequal to $p$). Thanks to computers, brute force searches have shown that the conjecture holds true as high as $n = 10^5$. % I look forward to updates. Mathematica confirms to ten thousand in 17 seconds, I'm still waiting for an answer on one million