A Proof of Legendre's Conjecture

Question

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?

Answer 1

The failure is precisely at the last line on the next-to-last page. Remember:

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

Answer 2

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 $n$.