The Fermat-Catalan conjecture, in number theory, generalizes both Fermat's Last Theorem and Catalan's conjecture. It states that the equation a to the power m plus b to the power n equals c to the power k, where a, b and c are positive coprime integers and m, n and k are positive integers satisfying one over m plus one over n plus one over k less than 1, has only finitely many solutions with distinct triples of values. The inequality on the exponents is essential, since without it there would be infinitely many solutions.
Facts
StatementThe Fermat-Catalan conjecture, generalizing Fermat's Last Theorem and Catalan's conjecture, states that the equation a to the power m plus b to the power n equals c to the power k, where a, b and c are positive coprime integers and m, n and k are positive integers with one over m plus one over n plus one over k less than 1, has only finitely many solutions with distinct triples a to the m, b to the n, c to the k. 1 Progress Toward ResolutionBy the Darmon-Granville theorem, which uses Faltings' theorem, only finitely many coprime solution triples exist for any single fixed choice of exponents m, n and k satisfying the inequality, though the conjecture across all admissible exponent choices remains open; it is known that the abc conjecture would imply the Fermat-Catalan conjecture. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Fermat-Catalan Conjecture (Wikipedia)
Sources
1. Fermat-Catalan Conjecture (Wikipedia)
Lead section
the Fermat-Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture
Progress section
It is known by the Darmon-Granville theorem, which uses Faltings' theorem, that for any fixed choice of positive integers m, n and k satisfying (2), only finitely many coprime triples (a, b, c) solving (1) exist.
- In Branch: Number Theory, Lead sentence
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.