## Set Theory and Logic   ## Elementary Set Theory Used To Prove FLT

Authors: Phil A. Bloom

An open problem is proving FLT \emph{simply} (using Fermat's toolbox) for each $n\in\mathbb{N}, n>2$. Our \emph{direct proof} (not BWOC) of FLT is based on our algebraic identity $((r+2q^n)^\frac{1}{n})^n-((r-2q^n)^\frac{1}{n})^n=(2^\frac{2}{n}q)^n$ with arbitrary values of $n\in\mathbb{N}$, and with $r\in\mathbb{R},q\in\mathbb{Q},n,q,r>0$. For convenience, we \emph{denote} $(r+2q^n)^\frac{1}{n}$ by $s$; we \emph{denote} $(r-2q^n)^\frac{1}{n}$ by $t$; and, we \emph{denote} $2^\frac{2}{n}q$ by $u$. For any given $n>2$ : Since the term $u$ or $2^\frac{2}{n}q$ with $q\in\mathbb{Q}$ is not rational, this identity allows us to relate null sets $\{(s,t,u)|s,t,u\in\mathbb{N},s,t,u>0,s^n-t^n=u^n\}$ with subsequently proven null sets $\{z,y,x|z,y,x\in\mathbb{N},z,y,x>0,z^n-y^n=x^n\}$. We show it is true, for $n>0$, that $\{u|s,t,u\in\mathbb{N},s,t,u>0,s^n-t^n=u^n\}=\{x|z,y,x\in\mathbb{N},z,y,x>0,z^n-y^n=x^n\}$. Hence, for any given $n\in\mathbb{N},n>2$, it is a true statement that $\{(x,y,z)|x,y,z\in\mathbb{N},x,y,z>0,x^n+y^n=z^n\}=\varnothing$.

### Submission history

[v1] 2018-11-01 11:11:48
[v2] 2018-11-05 17:07:33
[v3] 2018-11-06 14:00:16
[v4] 2018-11-09 17:10:10
[v5] 2018-11-25 17:58:44
[v6] 2018-12-09 22:58:42
[v7] 2019-01-10 11:46:28
[v8] 2019-01-15 22:07:17
[v9] 2019-01-21 17:31:05
[vA] 2019-01-26 23:01:07
[vB] 2019-02-25 18:13:56
[vC] 2019-03-11 17:10:11
[vD] 2019-03-17 21:55:04