The inter-universal Teichm\"uller 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 Teichm\"uller 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