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

Abstract

[This is an older version of the paper, which will be updated soon.] In the present paper, we continue our research on the generalized Fermat equation $x^r+y^s=z^t$ with signature $(r,s,t)$, where $r,s,t\ge 2$ are positive integers such that $\frac{1}{r}+\frac{1}{s}+\frac{1}{t}<1$. All known positive primitive solutions for the generalized Fermat equation when $\frac{1}{r}+\frac{1}{s}+\frac{1}{t}<1$ are related to the Catalan solutions $1^n+2^3=3^2$ and nine non-Catalan solutions. By applying inter-universal Teichmüller theory and its slight modification in the case of elliptic curves over rational numbers, we deduce that the generalized Fermat equation $x^r+y^s=z^t$ has no non-trivial primitive solution except for those related to the Catalan solutions and nine non-Catalan solutions mentioned above, when $(r,s,t)$ is not a permutation of the following signatures: $\bullet$ $(4,5,n)$, $(4,7,n)$, $(5,6,n)$, with $7\le n\le 303$. $\bullet$ $(2,3,n)$, $(3,4,n)$, $(3,8,n)$, $(3,10,n)$, with $11\le n\le 109$ or $n\in \{113,121\}$. $\bullet$ $(3,5,n)$, with $7\le n\le 3677$; $(3,7,n)$, $(3,11,n)$, with $11\le n\le 667$. $\bullet$ $(3,m,n)$, with $13\le m\le 17$, $m<n\le 29$; $(2,m,n)$, with $m\ge 5$, $n\ge 7$. As a corollary, to solve the generalized Fermat equation $x^r+y^s=z^t$ with exponents $r,s,t\ge 4$, we are left with $244$ signatures $(r,s,t)$ up to permutation; to solve the Beal conjecture, we are left with $2446$ signatures $(r,s,t)$ up to permutatio