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http://vixra.org/abs/1303.0048

In the paper the author proposes an elementary proof of Legendre's Conjecture. I was wondering if the proof is correct, because till now, there is no accepted proof of Legendre's Conjecture. On one first glance the proof seemed correct, but there may be some subtle mistake that I am unable to detect.

Is the proof correct?

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closed as off-topic by Stella Biderman, Henning Makholm, Adrian Keister, Namaste, Xander Henderson Aug 19 '18 at 0:47

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    $\begingroup$ The link does not work for me. $\endgroup$ – Clarinetist Jun 25 '14 at 4:21
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    $\begingroup$ Not to be snide, but the paper is badly written and cites a grand total of one reference (which is a survey text rather than a research paper), which are usually not good signs. $\endgroup$ – anomaly Jun 25 '14 at 6:44
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    $\begingroup$ @anomaly Publishing on viXra is already a bad sign, it seems: en.wikipedia.org/wiki/ViXra. And when the author presents himself as "World order Number Theorist", it's really a bad sign. $\endgroup$ – Jean-Claude Arbaut Sep 24 '14 at 7:59
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    $\begingroup$ @Jean-ClaudeArbaut: Yeah, viXra is the equivalent of vanity publishing for math and science. It's for people who totally have an elementary, five-page proof of the Riemann Hypothesis, Fermat's Last Theorem, Legendre's Conjecture, etc. and need to announce it to the world. Even if one lacks the academic credentials to get an arxiv account, there are much better options available. $\endgroup$ – anomaly Sep 24 '14 at 10:03
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    $\begingroup$ I'm voting to close this question as off-topic because checking the validity of arXiv articles is not within the scope of this site. $\endgroup$ – Stella Biderman Aug 18 '18 at 18:07
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The failure is precisely at the last line on the next-to-last page. Remember:

If $a\leq b$, then $-a\geq-b$. Not $-a\leq -b$.

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    $\begingroup$ Was it that easy? My god! I actually did not checked the complete calculation because I thought that an author trying to prove Legendre's Conjecture will not make silly mistakes. $\endgroup$ – William Hilbert Jun 25 '14 at 12:06
  • $\begingroup$ @WilliamHilbert there are far more cranks who think they’re brilliant mathematicians than you’d ever dream. $\endgroup$ – Stella Biderman Aug 18 '18 at 18:08
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I'm not willing to slog through all that computation, but the punchline of the paper is the assertion that $\pi((n+1)^2) - \pi(n^2) \geq \pi(2n) - \pi(n) > 0$ for $n\geq 5$, the latter inequality coming from Bertrand's postulate. The first inequality is false, however; it fails (according to Mathematica, at least) for $n = 42$ and quite a few other $n$.

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