Doron Zeilberger suggested the following potential proof for Fermat's last theorem:
Let's define: $$W(n,a,b,c) \equiv (a^n + b^n - c^n)^2$$ I am almost sure that there exists a polynomial, discoverable by computer, with positive coefficients such that: $$W(n,a,b,c) = P\left(W(n,a-1,b,c), W(n,a,b-1,c), \ldots W(n -1,a,b,c), \ldots\right)$$ for $n>3$.
Since $W > 0$ for $n= 3$, and $abc>0$ FLT would follow.
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Could someone explain how / why exactly "FLT would follow"?
Moreover, why wouldn't one have to find a separate polynomial for each (unbound) $n$?
The proof is by infinite descent. Let $n,a,b,c$ be the smallest possible solution to $W(n,a,b,c) = 0$, where $n>3$. Since he is almost sure that there exists a polynomial, discoverable by computer, with positive coefficients such that: $$W(n,a,b,c) = P\left(W(n,a-1,b,c), W(n,a,b-1,c), \ldots W(n -1,a,b,c), \ldots\right)$$ for $n>3$. This would mean that $W(n,a,b,c) > 0$ contradicting the fact that $a,b,c$ is the smallest possible solution to $W(n,a,b,c) = 0$.