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>
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:<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'',&nbsp;''b'',&nbsp;''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}}

Latest revision as of 14:04, 11 December 2014


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