José Cayolla: Fermat's Last Theorem admits an infinity of proving ways and two corollaries. arXiv:1507.06989 [math.GM]
I don't usually devote so much time to "crackpot papers", but I have a friend who thinks there may be something to this one and has written up the complete proof a few different ways in order to make it as clear as possible to me, and so I feel the need to pinpoint exactly why this paper is wrong. Unfortunately neither the original author nor my friend is a native English speaker, and I don't want to let trivial errors get in the way of finding the real problems here.
To summarize the argument, we want to show that $x^n+y^n=z^n$ has no solutions for $x<y<z$ and $n\ge3$, and we view this as fixing $x,y,z$ and varying $n$. For small $n$ we have $z^n<x^n+y^n$ and for large $n$ this reverses to $z^n>x^n+y^n$, and there is at most one $n$ for which equality is attained (which would be a counterexample). Letting $n-1$ be the largest number satisfying $z^{n-1}<x^{n-1}+y^{n-1}$, FLT is equivalent to showing $z^n>x^n+y^n$.
(page 7:) The "infinity of proving ways" in the title refers to the interval $z^n\ge \zeta_n\ge x^n+y^n$ of possible witnesses $\zeta_n$ to this inequality. (Here I am a bit concerned: why not $z^n>\zeta_n\ge x^n+y^n$?) To find these $\zeta$ values we use another auxiliary $\lambda$ which satisfies the equation $$\lambda z^{n-1}\geq x^{n-1}+y^{n-1}\implies z^n>x^n+y^n.\tag{19}$$ (The text before this equation seems to suggest $\lambda z^{n-1}\geq x^{n-1}+y^{n-1}\wedge z^n>x^n+y^n$ instead, but perhaps I am misreading.) I think there is an error near $(22)$ and $(23)$, and it seems like the meat of the argument is between page 7 and page 9, where the text claims "this FLT full proof is now completed". Even earlier, $(27)$ is establishing the inequality that implies FLT at the beginning of page 8.
I've done my best to narrow the focus of this question to a particular part of the text where I think the error is in order to limit the scope, but this is a text verification question, and the best kind of answer would explain why the author's approach can't work even after fixing trivial errors.