## Last Fermat Theorem, sumation, a couple of updates on u-function

By LFT, $z^n-y^n=x^n$, for $x,y,z \in N$, ie. $x^n=\Sigma_{i=0}^{i=n}{y^i z^{n-i}}$. When $S(n)=\Sigma_{i=0}^{i=n}{y^i z^{n-i}}$, then $S(n)-xyS(n-2)=x^n+y^n$. From there, we have $x^n=(z-y)S_{n-1}$ and $S_n=z^n+y^n+zyS(n-2)$, which leads to $S_n=(y+{{a}\over{S_{n-1}}})^n+y^n+y(y+{{a}\over{S_{n-1}}})S_{n-2}$ for a fixed $a \in N$, where $a$ is $n$-th square of a chosen integer, thus $u(a)>7$. From there, we have $S(n)-xyS(n-2)=z^n$. Haven’t gone further with this yet, as more in-depth analysis of the very sumations is required.

Another interesting thing, for $n=3$ of the LFT, is that from $z^3=(z-y)(z^2+zy+y^2)$ and $a={{p}\over{q}}$ for $p,q \in N$, where it is trivial to show that $a<1$, with substitutions $z=y+ax$ and $z^2+zy+y^2=x^2a^{-1}$, we have determinant where ${{3p^3}\over{q^3}}$ can’t must never be integer for contradiction to hold, assuming no leaks in analysis. As for the substitutions $z-y={{x}\over{k}}$ and $z^2+zy+y^2=kx^2$ for some $k\in N$, we must make sure so that generality is not affected. Wanted to use this transformation as some sort (long-vision) of equivalence between the elementary NT and ring homomorphism (and its numerical criterion) from Wiles .

When it comes to $u$-function, it’s trivial to show that for $a,b \in N, u(b)>u(a)$ either $u(a)$ nor $u(b)$ are not the borderlines for $u(a+b)$, so the questions that arise are: is $u(a+b)$  limited by either the amount of primes equal to the argument, or – for instance – the smallest prime appearing in the sum equal to the very argument. I will update you on this shortly as soon as I have more information.

Yet another update on learning.