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I have read and heard conflicting reports about whether or not Legendre's conjecture has been proven. Refresh: Legendre believed that there will always be at least one prime between $n^2$ and $(n+1)^2$.

Most websites have told me that it remains unproven, however a few claim that it has. The main article is: http://vixra.org/pdf/1303.0048v1.pdf

The article was written sometime last year and claims to have a proof but many other sites updated recently say it remains unsolved.

So, is it solved or unsolved?

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    $\begingroup$ As a rule of thumb, don't trust things posted at vixra. $\endgroup$
    – Pedro
    Commented Apr 19, 2014 at 2:43
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    $\begingroup$ That would represent an enormous leap in our knowledge about prime gaps. Skepticism is warranted. $\endgroup$ Commented Apr 19, 2014 at 3:08
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    $\begingroup$ I'd say extreme skepticism is warranted. $\endgroup$ Commented Apr 27, 2014 at 3:00

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I know this is an older question, but this was also asked here: A Proof of Legendre's Conjecture where Gina points out there is an error in one of the inequalities, rendering the proof invalid.

In short, yes, Legendre's Conjecture remains unresolved.

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