The Beal conjecture for generalized Fermat equations

About 5 years old · traced to 2021

Consider the generalized Fermat equation

x^p+y^q=z^r.

A solution $(a,b,c)\in\mathbb{Z}^3$ is primitive if $\operatorname{gcd}(a,b,c)=1$ and trivial if $abc=0$ . The Beal Prize conjecture. If $p,q,r\geq 3$ , then there are no non-trivial primitive solutions to $x^p+y^q=z^r$ . This conjecture is attributed in the source to Beal, Granville, Tijdeman, and Zagier, and concerns the existence of non-trivial primitive solutions to generalized Fermat equations. It remains open; the source notes that the ABC conjecture would imply only finitely many counterexamples.

References

Primary source

Imin Chen and Angelos Koutsianas, “Darmon's Program: A survey”, arXiv:2507.15149 (2025).

Additional references

3 papers in this index state this conjecture (2021–2025). The statement above is taken from the most recent of them; the others are arXiv:2206.04290, arXiv:2104.04522.

diophantine-equations generalized-fermat-equations

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A purported proof was posted in 2024, but no independent verification has established the conjecture.

The conjecture says that, for exponents at least three, the equation $x^p+y^q=z^r$ has no non-trivial primitive integer solution. It is attributed to Beal, Granville, Tijdeman, and Zagier, and remains unresolved in the verified literature.

Known results

Darmon and Granville (1995): for fixed exponents with $1/p+1/q+1/r<1$ , only finitely many primitive solutions exist.
Darmon and Merel: no coprime solutions occur with signature $(p,p,3)$ for $p\geq 3$ .
A 2024 survey reports that the known Fermat–Catalan solutions all have at least one exponent equal to $2$ .
Special infinite families, including signatures $(r,r,p)$ with selected coefficients, have been solved in some cases.

2024 claimed proofs

On May 19, 2024, a work titled The Proof of the Beal Conjecture claimed a reduction using “growth functions” and Fermat’s Last Theorem. Another undated work claimed analogous reductions for unequal and partially equal exponents. Neither claim has independent verification, referee confirmation, or accepted publication establishing the conjecture.

Current status (as of August 2026): The conjecture remains unverified and open; substantial fixed-exponent and special-signature results are known, while the 2024 purported proofs remain unconfirmed.

Sources

No solutions have been posted yet.