Is this elementary proof of FLT correct? Consider the classic FLT for $n=3$, $x^3+y^3=z^3 $ 
Without loss of generality,we can rewrite as $a^3+(b-a)^3=c^3…Equation (1)$
We can also assume there is no common factor between $a,b-a$ and $c …Assumption (1)$
Since $m+n$ is a factor of $m^3+n^3$, $b$ divides $c^3$.
Let $p$ be a common prime factor of $b$ and $c … Assumption (2)$
$p$ does not divide $a$ as if it did, $p$ divides $b-a$ and hence $p$ divides $a$ violating $Assumption (1)$
Rewriting Equation(1)  we get, $a^3+b^3-a^3-3b^2 a+3ba^2=c^3…Equation (2)$
rearranging we get, $b^3-c^3=3b^2 a-3ba^2 …Equation (3)$
Note that $p^3$ divides the LHS of $Equation (3)$
Two cases: 
Case I: $p \ne $n
On the RHS of $Equation (3)$, $p^2$ divides the first term $3b^2 a$ but not $3ba^2$
Case II: $p = $n
Now $p^3$ divides $3b^2a$ but not $3ba^2$.
In both cases, LHS of $Equation (3) \ne RHS $
The General case of FLT for any odd prime n:
Without loss of generality,we can rewrite FLT as $a^n+(b-a)^n=c^n…Equation (4)$
Expanding, we get
$a^n+b^n-nC_1 b^{n-1} a+nC_2 b^{n-2} a^2-⋯-nC_{n-2} b^2 a^{n-2}+nC_{n-1} ba^{n-1}-a^n=c^n…Equation (5)$
rearranging we get,
$b^n-c^n=nC_1 b^{n-1} a- nC_2 b^{n-2} a^2+ ⋯+nC_{n-2} b^2 a^{n-2}-nC_{n-1} ba^{n-1}…Equation (6)$
Therefore $p^n$  divides the LHS of $Equation (6)$
Two cases.Case I: $p \ne n$
On RHS of Equation (6), $p^2$ divides all the terms except the last term $nC_{n-1} ba^{n-1}$
Case II: $p=n$
Now $p^3$ divides all terms of RHS except the last term $nC_{n-1} ba^{n-1}$
In both cases, LHS of $Equation (6) \ne RHS $

The proof is incorrect. The error is in the statement
$$``\text{$p^2$ divides the first term $3b^{2}a$ but not $3a^{2}b$}."$$
In case 1, when $p$ is different than $3$, you know that $p \nmid a$, $p \nmid 3$, and $p\mid b$. But, you do not know that $p^{2} \nmid b$. This is because you chose $p$ to be any prime factor of $b$ and $c$ (assumption 2), which does not exclude the possibility that $p^{2} \mid b$.
So, if $p^{2} \mid b$, then the claim that $p^{2} \nmid 3a^{2}b$ is false, and the conclusion no longer follows.
