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- Fermat–Catalan_conjecture abstract "In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation</dd></dl>has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck); here a, b, c are positive coprime integers and m, n, k are positive integers satisfying</dd></dl>As of 2008, the following solutions to (1) are known:The first of these (1m+23=32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since we can pick any m for m>6), these solutions only give a single triplet of values (am, bn, ck).It is known by 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, but of course the full Fermat–Catalan conjecture is a much stronger statement since it allows for an infinitude of sets of exponents m, n and k.The abc conjecture implies the Fermat–Catalan conjecture.Beal's conjecture is that all FCC solutions use 2 as an exponent.".
- Fermat–Catalan_conjecture abstract "In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equationhas only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck); here a, b, c are positive coprime integers and m, n, k are positive integers satisfyingAs of 2008, the following solutions to (1) are known:The first of these (1m+23=32) is the only solution where one of a, b or c is 1, according to the Catalan conjecture, proven in 2002 by Preda Mihăilescu. While this case leads to infinitely many solutions of (1) (since we can pick any m for m>6), these solutions only give a single triplet of values (am, bn, ck).It is known by 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, but of course the full Fermat–Catalan conjecture is a much stronger statement since it allows for an infinitude of sets of exponents m, n and k.The abc conjecture implies the Fermat–Catalan conjecture.Beal's conjecture is that all FCC solutions use 2 as an exponent.".
- Fermat–Catalan_conjecture wikiPageID "22736152".
- Fermat–Catalan_conjecture wikiPageRevisionID "574935105".
- Fermat–Catalan_conjecture subject Category:Conjectures.
- Fermat–Catalan_conjecture subject Category:Number_theory.
- Fermat–Catalan_conjecture comment "In number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name.".
- Fermat–Catalan_conjecture label "Conjetura de Fermat–Catalan".
- Fermat–Catalan_conjecture label "Fermat–Catalan conjecture".
- Fermat–Catalan_conjecture label "フェルマー=カタラン予想".
- Fermat–Catalan_conjecture sameAs Fermat%E2%80%93Catalan_conjecture.
- Fermat–Catalan_conjecture sameAs Fermatova-Catalanova_domněnka.
- Fermat–Catalan_conjecture sameAs Conjetura_de_Fermat–Catalan.
- Fermat–Catalan_conjecture sameAs フェルマー=カタラン予想.
- Fermat–Catalan_conjecture sameAs Q851604.
- Fermat–Catalan_conjecture sameAs Q851604.
- Fermat–Catalan_conjecture wasDerivedFrom Fermat–Catalan_conjecture?oldid=574935105.