Dice Question Streamline Icon: https://streamlinehq.com

Fermat–Catalan Conjecture (Integers)

Establish that the Diophantine equation a^p + b^q = c^r admits only finitely many solutions (a, b, c, p, q, r) in positive integers satisfying 1/p + 1/q + 1/r < 1.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors present the Fermat–Catalan conjecture as a classical number-theoretic conjecture generalizing Fermat’s Last Theorem, and then prove a polynomial analogue using Mason–Stothers. Including this conjecture clarifies the motivation for their polynomial results and highlights the contrast between the open integer case and the established function-field case.

Their polynomial variant (Theorem 2.1) shows nonexistence of nonconstant polynomial solutions under analogous conditions, underscoring how Mason–Stothers provides a successful function-field analogue where the integer case remains conjectural.

References

The Fermat--Catalan conjecture is a generalization of Fermat's Last Theorem stating that the equation $ap + bq = cr$ has only finitely many solutions $(a, b, c, p, q, r)$ in positive integers satisfying $1/p + 1/q + 1/r < 1$ .

— Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4 (2408.15180 - Baek et al., 27 Aug 2024) in Section 2, Statements of the Theorem and its Corollaries (paragraph before Theorem 2.1)