The inter-universal Teichmüller theory and new Diophantine results over rational numbers. II
Zhong-Peng Zhou
TL;DR
This work extends inter-universal Teichmüller theory (IUT) to the generalized Fermat equation $x^r+y^s=z^t$ with signatures $(r,s,t)$ satisfying $\frac{1}{r}+\frac{1}{s}+\frac{1}{t}<1$, establishing that all positive primitive solutions are Catalan-related or among nine known non-Catalan examples, except for permutations belonging to explicitly listed signature families. It introduces a $2$-torsion variant of IUT theory, constructs initial $\Theta$-data via Frey-curve analyses, and develops a log-volume local-global framework to derive computable upper bounds on the j-invariant denominator $N$ and related ramification data, enabling finite reductions across broad signature classes. The paper treats the general $(r,s,t)$ case and the particular families $(2,3,t)$ and $(3,r,s)$, delivering explicit upper bounds on exponents and strong structural restrictions that reduce the solution search to a finite computation and show no new primitive solutions beyond the Catalan/known nine. Consequently, solving Beal reduces to verifying $2446$ signatures up to permutation, and the overall generalized Fermat problem to $244$ signatures for $r,s,t\ge4$, with Catalan bounds playing a crucial role in constraining exponents and guiding computational verification.
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