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 (|abc|) 700, we prove |abc| 3 (abc) + 8|abc| |abc|. This implies for any 0 < 110, |abc| \ (400 ^{-2 (^-1) ), \ rad (abc) ^3+3\}, reducing the constant in effective abc bounds from 1. 7 10^30 (Mochizuki-Fesenko-Hoshi-Minamide-Porowski) to 400. For positive primitive solutions (x, y, z) to the generalized Fermat equation xʳ + yˢ = zᵗ (r, s, t 3), define h = (xʳ yˢ zᵗ). We prove explicit bounds: gather* h 573\ \ (r, s, t 8) ; \; h 907\ \ (r, s, t 5) ; \; h 2283\ \ (r, s, t 4) ; \\ h 14750\ \ (\r, s\ 4\ or\ t 4) ; \; h 24626\ \ (r, s, t 3). gather* These imply Fermat's Last Theorem (FLT) holds unconditionally for prime exponents 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 20 or (r, s, t) is a permutation of (3, 3, n) (n 3).
Ze‐Hua Zhou (Sat,) studied this question.