{{Short description|Generalization of Fermat's Last Theorem and of Catalan's conjecture,}} {{Use dmy dates|date=March 2024}} In number theory, the '''Fermat–Catalan conjecture''' is a generalization of Fermat's Last Theorem and of Catalan's conjecture. 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>) where ''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}}}} 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'' = ''a''<sup>''m''</sup> + ''b''<sup>''n''</sup>), with ''m''=''n''=''k''=2 (for the infinitely many Pythagorean triples), and e.g. <math>7^5 + 393^3 = 7792^2</math>.
==Known solutions== As of 2024, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:<ref name="Sikora.2024">{{cite report | arxiv=2410.21552 | author=Adam S. Sikora | title=Fermat-Catalan and Tijdeman-Zagier Conjectures for Products | institution=ArXiv | type=Technical Report | number=2410.21552 | pages= | date=Oct 2024 }}</ref><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><ref>{{cite journal | doi=10.1215/S0012-7094-98-09105-0 | author=Frits Beukers | author-link=Frits Beukers |title=The Diophantine equation ''Ax<sup>p</sup>''+''By<sup>q</sup>''=''Cz<sup>r</sup>'' | journal=Duke Math. J. | volume=91 | number=1 | pages=61–88 | date=Jan 1998 }} Here: p.61: "the larger [solutions] were found by a computer search performed on Fermat day at Utrecht in November 1993 ... Notice that in each solution an exponent 2 occurs."</ref>
:<math>1^m + 2^3 = 3^2\;</math> (for <math>m>6</math> to satisfy Eq. 2) :<math>2^5 + 7^2 = 3^4\;</math> :<math>7^3 + 13^2 = 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 one 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>).
==Partial results== 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.<ref>{{cite journal |first1=H. |last1=Darmon |first2=A. |last2=Granville |title=On the equations ''z''<sup>''m''</sup> = ''F''(''x'', ''y'') and ''Ax''<sup>''p''</sup> + ''By''<sup>''q''</sup> = ''Cz''<sup>''r''</sup> |journal=Bulletin of the London Mathematical Society |volume=27 |pages=513–43 |year=1995 |issue=6 |doi=10.1112/blms/27.6.513 |doi-access=free }}</ref><ref name=Elkies>{{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}}</ref>{{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.<ref>{{cite book | last = Waldschmidt | first = Michel | authorlink = Michel Waldschmidt | contribution = Lecture on the <math>abc</math> 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| isbn = 978-3-0348-0858-3 }}</ref>
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.
Poonen et al.<ref>{{cite report | arxiv=math/0508174 | first1=Bjorn | last1=Poonen | first2=Edward F. | last2=Schaefer | first3=Michael | last3=Stoll | title=Twists of ''X''(7) and primitive solutions to ''x''<sup>2</sup> + ''y''<sup>3</sup> = ''z''<sup>7</sup> | date=Aug 2005 }} Here: p.3, table 1.</ref><ref name="Poonen.Schaefer.Stoll.2007">{{cite journal | first1=Bjorn | last1=Poonen | first2=Edward F. | last2=Schaefer | first3=Michael | last3=Stoll | title=Twists of ''X''(7) and primitive solutions to ''x''<sup>2</sup> + ''y''<sup>3</sup> = ''z''<sup>7</sup> | journal=Duke Math. J. | volume=137 | pages=103–158 | year=2007 }}</ref> list exponent triples where the solutions have been determined:<ref group=note>The notation "{''p'',''q'',''r''}" means that the solutions have been determined for every permutation of (''p'',''q'',''r'').</ref> {2,3,7},<ref name="Poonen.Schaefer.Stoll.2007"/> {2,3,8},<ref name="Bruin.1999">{{cite journal | mr=1711307 | url=https://www.cambridge.org/core/journals/compositio-mathematica/article/diophantine-equations-x2-y4z6-and-x2y8-z3/A1F11FEADBCFE720756819531A5E9CAE | author=Nils Bruin | title=The Diophantine Equations ''x''<sup>2</sup> ± ''y''<sup>4</sup> = ± ''z''<sup>6</sup> and ''x''<sup>2</sup> + ''y''<sup>8</sup> = ''z''<sup>3</sup> | journal=Compositio Mathematica | volume=118 | number= | pages=305–321 | year=1999 }}</ref><ref name="Bruin.2003">{{cite journal | mr=2011330 | url=https://www.cecm.sfu.ca/~nbruin/bruin_crelle.pdf | author=Nils Bruin | title=Chabauty methods using elliptic curves | journal=J. Reine Angew. Math. | volume=562 | number= | pages=27–49 | year=2003 }}</ref> {2,3,9},<ref>{{cite journal | mr=2047096 | url=https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01633-3/S0025-5718-04-01633-3.pdf | author=Nils Bruin | title=Visualising ''Sha[2]'' in abelian surfaces | journal=Math. Comp. (electronic) | volume=73 | number= | pages=1459–1476 | year=2004 }}</ref> {2,2''q'',3} for prime 7<''q''<1000 with ''q''≠31,<ref>{{cite journal | url= | author=Imin Chen | title=On the equation ''s''<sup>2</sup> + ''y''<sup>2''p''</sup> = ''α''<sup>3</sup> | journal=Math. Comp. | volume=77 | number=262 | pages=1223–1227 | year=2007 }}</ref> {2,4,5},<ref name="Bruin.2003"/> {2,4,6},<ref name="Bruin.1999"/> (2,4,7), (2,4,''q'') for prime ''q''≥211,<ref>{{cite journal | mr=2075481 | jstor=40067960 | url=https://people.math.wisc.edu/~ellenberg/A4B2Cp.pdf | author=Jordan S. Ellenberg | title=Galois representations attached to ''Q''-curves and the generalized Fermat equation ''A''<sup>4</sup> + ''B''<sup>2</sup> = ''C''<sup>''p''</sup> | journal=Amer. J. Math. | volume=126 | number= | pages=763–787 | year=2004 }}</ref> (2,''n'',4),<ref>"follows easily from" {{cite journal | mr=2031121 | url=https://personal.math.ubc.ca/~bennett/BS.pdf | author=Michael A. Bennett and Chris M. Skinner | title=Ternary Diophantine equations via Galois representations and modular forms | journal=Canad. J. Math. | volume=56 | number=1 | pages=23–54 | year=2004 }}</ref><ref>the special case (4,''n'',4) was done earlier in {{cite journal | mr=1260076 | url=https://www.math.mcgill.ca/darmon/pub/Articles/Research/10.Fermat-44p/paper.pdf | author=Henri Darmon | title=The equation ''x''<sup>4</sup> - ''y''<sup>4</sup> = ''z''<sup>p</sup> | journal=C. R. Math. Rep. Acad. Sci. Canada | volume=15 | number= | pages=286–290 | year=1993 }}</ref> {2,''n'',''n''},<ref name="Darmon.Merel.1997">{{cite journal | mr=1468926 | url=https://eudml.org/doc/153941 | author=Henri Darmon and Loc Merel | title=Winding quotients and some variants of Fermats last theorem | journal=J. Reine Angew. Math. | volume=490 | number= | pages=81–100 | year=1997 }}</ref> {3,3,4},<ref name="Bruin.2000">{{cite book | mr=1850605 | url=https://www.cecm.sfu.ca/~nbruin/eq33p.ps.gz | author=Nils Bruin | contribution=On powers as sums of two cubes | pages=169–184 |editor=Wieb Bosma | title=Algorithmic Number Theory – 4th Intnl. Symp. ANTS | location=Leiden | publisher=Springer | series=LNCS | volume=1838 | year=2000 }}</ref> {3,3,5},<ref name="Bruin.2000"/> {3,3,''q''} for 17≤''q''≤10000,<ref>{{cite journal | mr=1618290 | url = https://projecteuclid.org/journals/experimental-mathematics/volume-7/issue-1/Sur-l%C3%A9quation-asp-3bsp-3csp-p/em/1047674269.pdf | author=Alain Kraus | title=Sur l'équation ''a''<sup>3</sup> + ''b''<sup>3</sup> = ''c''<sup>''p''</sup> | journal=Experiment. Math. | volume=7 | number= | pages=1–13 | year=1998 }}</ref> {3,''n'',''n''},<ref name="Darmon.Merel.1997"/> {2''n'',2''n'',5},<ref>{{cite journal | url=https://personal.math.ubc.ca/~bennett/bennetjtnb.pdf | author=Michael A. Bennett | title=The equation ''x''<sup>2''n''</sup> + ''y''<sup>2''n''</sup> = ''z''<sup>5</sup> | journal=Journal de Théorie des Nombres de Bordeaux | volume=18 | number= | pages=315–321 | year=2006 }}</ref> {''n'',''n'',''n''}.<ref>{{cite journal | author=Andrew Wiles | title=Modular Elliptic Curves and Fermat's Last Theorem | journal=Annals of Mathematics | volume=142 | pages=443–551 | year=1995 }}</ref><ref>{{cite journal | mr=1333036 | url=https://staff.fnwi.uva.nl/a.l.kret/Galoistheorie/taylor-wiles.pdf | author=Richard Taylor and Andrew Wiles | title=Ring-theoretic properties of certain Hecke algebras | journal=Ann. of Math. (2) | volume=141 | pages=553–572 | year=1995 }}</ref> For each of these exponent triples, if there is some solution at all, it is listed among those in section {{section link|#Known solutions}}.
Sikora partially used the cluster computers at the Center for Computational Research at University at Buffalo to test all tuples (''a'',''b'',''c'',''m'',''n'',''k'') such that min(''m'',''n'',''k'') ≤ 113 and ''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>, ''c''<sup>''k''</sup> < ''M''<sub>min(''m'',''n'',''k'')</sub>, where ''M''<sub>2</sub> = 2<sup>71</sup>, ''M''<sub>3</sub> = 2<sup>80</sup>, ''M''<sub>4</sub> = 2<sup>100</sup>, and ''M''<sub>5</sub> = ... = ''M''<sub>113</sub> = 2<sup>113</sup>. He did not find any other solution than those above.<ref group=note>For example, the five known large solutions were all reproduced during the test for min(''m'',''n'',''k'') = 2, where ''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>, and ''c''<sup>''k''</sup> were considered up to 2<sup>71</sup>.</ref><ref name="Sikora.2024"/>
==See also==
*Sums of powers, a list of related conjectures and theorems
==Notes== {{reflist|group=note}}
==References== {{reflist}}
==External links==
* [http://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/PerfectPowers.pdf ''Perfect Powers: Pillai's works and their developments''. Waldschmidt, M. ] * {{Cite OEIS|A214618|Perfect powers z^r that can be written in the form x^p + y^q, where x, y, z are positive coprime integers and p, q, r are positive integers satisfying 1/p + 1/q + 1/r < 1}}
{{DEFAULTSORT:Fermat-Catalan conjecture}} Category:Conjectures Category:Unsolved problems in number theory Category:Diophantine equations Category:Abc conjecture