The inter-universal Teichm\"uller theory and new Diophantine results over the rational numbers. I

Abstract

By applying inter-universal Teichm\"uller 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 $\log(|abc|) \geq 700$, we prove $$ \log|abc| \leq 3\log\mathrm{rad}(abc) + 8\sqrt{\log|abc| \cdot \log\log|abc|}. $$ This implies for any $0 < \epsilon \leq \frac{1}{10}$, $$ |abc| \leq \max\left\{\exp\left(400 \cdot \epsilon^{-2} \cdot \log(\epsilon^{-1})\right),\ \mathrm{rad}(abc)^{3+3\epsilon}\right\}, $$ 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 $x^r + y^s = z^t$ ($r, s, t \geq 3$), 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 $\geq 11$. 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 $r, s, t \geq 20$ or $(r, s, t)$ is a permutation of $(3, 3, n)$ ($n \geq 3$).