词条 | Fermat–Catalan conjecture |
释义 |
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 {{NumBlk|::||{{EquationRef|1}}}}has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m, n, k are positive integers satisfying {{NumBlk|::||{{EquationRef|2}}}}The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am + bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples). Known solutionsAs of 2015, the following ten solutions to (1) are known:[1] 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 one can pick any m for m > 6), these solutions only give a single triplet of values (am, bn, ck). Partial resultsIt is known by the Darmon–Granville theorem, which uses Faltings's 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.[2][3]{{rp|p. 64}} However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents m, n and k to vary. The abc conjecture implies the Fermat–Catalan conjecture.[4] For a list of results for impossible combinations of exponents, see Beal conjecture#Partial results. Beal's conjecture is true if and only if all Fermat–Catalan solutions have m = 2, n = 2, or k = 2. See also
References1. ^{{citation|first=Carl|last=Pomerance|authorlink=Carl Pomerance|contribution=Computational Number Theory|pages=361–362|title=The Princeton Companion to Mathematics|editor1-first=Timothy|editor1-last=Gowers|editor1-link=Timothy Gowers|editor2-first=June|editor2-last=Barrow-Green|editor3-first=Imre|editor3-last=Leader|editor3-link=Imre Leader|year=2008|publisher=Princeton University Press|isbn=978-0-691-11880-2}}. 2. ^{{cite journal |first1=H. |last1=Darmon |first2=A. |last2=Granville |title=On the equations zm = F(x, y) and Axp + Byq = Czr |journal=Bulletin of the London Mathematical Society |volume=27 |pages=513–43 |year=1995 |doi=10.1112/blms/27.6.513 }} 3. ^{{cite journal|last=Elkies| first = Noam D. | title=The ABC's of Number Theory | journal = The Harvard College Mathematics Review | year=2007 | volume=1 | issue = 1 | url=http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2}} 4. ^{{cite book | last = Waldschmidt | first = Michel | authorlink = Michel Waldschmidt | contribution = Lecture on the conjecture and some of its consequences | doi = 10.1007/978-3-0348-0859-0_13 | mr = 3298238 | pages = 211–230 | publisher = Springer | location = Basel | series = Springer Proc. Math. Stat. | title = Mathematics in the 21st century | url = http://www.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf | volume = 98 | year = 2015}} External links
2 : Conjectures|Diophantine equations |
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