Semigroup with two elements: Difference between revisions
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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|::|<math>a^m + b^n = c^k\quad</math>|{{EquationRef|1}}}} | |||
has only finitely many solutions (''a'',''b'',''c'',''m'',''n'',''k'') with distinct triplets of values (''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>, ''c''<sup>''k''</sup>); here ''a'', ''b'', ''c'' are positive [[coprime]] integers and ''m'', ''n'', ''k'' are positive integers satisfying | |||
{{NumBlk|::|<math>\frac{1}{m}+\frac{1}{n}+\frac{1}{k}<1.</math>|{{EquationRef|2}}}} | |||
As of 2008, the following solutions to (1) are known:<ref name=pcm>{{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}}.</ref> | |||
:<math>1^m+2^3=3^2\;</math> | |||
:<math>2^5+7^2=3^4\;</math> | |||
:<math>13^2+7^3=2^9\;</math> | |||
:<math>2^7+17^3=71^2\;</math> | |||
:<math>3^5+11^4=122^2\;</math> | |||
:<math>33^8+1549034^2=15613^3\;</math> | |||
:<math>1414^3+2213459^2=65^7\;</math> | |||
:<math>9262^3+15312283^2=113^7\;</math> | |||
:<math>17^7+76271^3=21063928^2\;</math> | |||
:<math>43^8+96222^3=30042907^2\;</math> | |||
The first of these (1<sup>''m''</sup>+2<sup>3</sup>=3<sup>2</sup>) 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 (''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>, ''c''<sup>''k''</sup>). | |||
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.<ref name=pcm/> | |||
[[Beal's conjecture]] is that all FCC solutions use 2 as an exponent. | |||
==See also== | |||
*[[Sums of powers]], a list of related conjectures and theorems | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Fermat-Catalan conjecture}} | |||
[[Category:Conjectures]] | |||
[[Category:Number theory]] | |||
{{numtheory-stub}} | |||
Revision as of 21:35, 6 October 2013
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 Template:NumBlk 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 Template:NumBlk
As of 2008, the following 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 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.[1]
Beal's conjecture is that all FCC solutions use 2 as an exponent.
See also
- Sums of powers, a list of related conjectures and theorems
References
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