The inter-universal Teichmüller theory and new Diophantine results over the rational numbers. I

Abstract

By applying inter-universal Teichmüller theory and its slight modification over the rational number field, we prove new Diophantine results towards effective abc inequalities and the generalized Fermat equations. For coprime integers $a,b,c$ satisfying $a+b=c$ and $\log (|abc|)\ge 700$, we prove $$\log |abc|\le 3\log \mathrm{rad}(abc)+8\sqrt{\log |abc|\cdot \log \log |abc|}.$$ This implies for any $0<\epsilon \le \frac{1}{10}$, $$|abc|\le \max \left\{\exp \left(400\cdot {\epsilon }^{-2}\cdot \log ({\epsilon }^{-1})\right),\mathrm{rad}(abc{)}^{3+3\epsilon }\right\},$$ reducing the constant in effective abc bounds from $1.7\cdot 10^{30}$ (Mochizuki-Fesenko-Hoshi-Minamide-Porowski) to $400$. For positive primitive solutions $(x,y,z)$ to the generalized Fermat equation $x^r+y^s=z^t$ ($r,s,t\ge 3$), define $h=\log (x^r{y}^s{z}^t)$. We prove explicit bounds: \begin{gather*} h \leq 573\ \ (r, s, t \geq 8); \; h \leq 907\ \ (r, s, t \geq 5); \; h \leq 2283\ \ (r, s, t \geq 4); \\ h \leq 14750\ \ (\min\{r, s\} \geq 4\ \text{or}\ t \geq 4); \; h \leq 24626\ \ (r, s, t \geq 3). \end{gather*} These imply Fermat's Last Theorem (FLT) holds unconditionally for prime exponents $\ge 11$. Combined with classical results for FLT with exponents $3,4,5,7$, this yields a new alternative proof of FLT. Computational verification confirms no non-trivial primitive solution exists when $r,s,t\ge 20$ or $(r,s,t)$ is a permutation of $(3,3,n)$ ($n\ge 3$).