Beal conjecture for the generalized Fermat equation

Prove that the generalized Fermat equation x^r + y^s = z^t has no positive primitive solutions when the exponents satisfy r, s, t ≥ 3; equivalently, show that there do not exist positive integers x, y, z with gcd(x, y, z) = 1 and integers r, s, t ≥ 3 such that x^r + y^s = z^t.

Background

The paper studies Diophantine inequalities and the generalized Fermat equation xr + ys = zt, deriving explicit bounds and non-existence results for several ranges of exponents. It surveys known results and solved signatures and highlights contexts where solutions are known or ruled out.

Within this survey, the authors explicitly recall Beal’s conjecture, which posits that the generalized Fermat equation admits no positive primitive solutions when all exponents are at least three. Although the paper establishes various upper bounds and non-existence results for certain signatures (e.g., large exponents and specific configurations), the full conjecture remains unresolved in general and is stated as such.

References

Since all known solutions have min{r, s, t} \le 2, Andrew Beal conjectured in 1993 (cf. ) that (\ref{GenFE}) has no positive primitive solution when r, s, t \ge 3. This conjecture is known as the Beal conjecture, also known as the Mauldin conjecture and the Tijdeman-Zagier conjecture.

— The inter-universal Teichmüller theory and new Diophantine results over the rational numbers. I  (2503.14510 - Zhou, 8 Mar 2025) in Section 4.1 (A brief survey)