Fermat–Catalan conjecture

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Short description: Generalization of Fermat's Last Theorem and of Catalan's conjecture,

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

am+bn=ck

 

 

 

 

(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

1m+1n+1k<1.

 

 

 

 

(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), with m=n=k=2 (for the infinitely many Pythagorean triples), and e.g. 75+3933=77922.

Known solutions

As of 2024, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:[1][2][3]

1m+23=32 (for m>6 to satisfy Eq. 2)
25+72=34
73+132=29
27+173=712
35+114=1222
338+15490342=156133
14143+22134592=657
92623+153122832=1137
177+762713=210639282
438+962223=300429072

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 results

It 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.[4][5]: 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.[6]

For a list of results for impossible combinations of exponents, see Beal conjecture. Beal's conjecture is true if and only if all Fermat–Catalan solutions have m = 2, n = 2, or k = 2.

Poonen et al.[7][8] list exponent triples where the solutions have been determined:[note 1]   {2,3,7},[8]   {2,3,8},[9][10]   {2,3,9},[11]   {2,2q,3} for prime 7<q<1000 with q≠31,[12]   {2,4,5},[10]   {2,4,6},[9]   (2,4,7),   (2,4,q) for prime q≥211,[13]   (2,n,4),[14][15]   {2,n,n},[16]   {3,3,4},[17]   {3,3,5},[17]   {3,3,q} for 17≤q≤10000,[18]   {3,n,n},[16]   {2n,2n,5},[19]   {n,n,n}.[20][21]   For each of these exponent triples, if there is some solution at all, it is listed among those in section § 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 am, bn, ck < Mmin(m,n,k), where M2 = 271, M3 = 280, M4 = 2100, and M5 = ... = M113 = 2113. He did not find any other solution than those above.[note 2][1]

See also

Notes

  1. ↑ The notation "{p,q,r}" means that the solutions have been determined for every permutation of (p,q,r).
  2. ↑ For example, the five known large solutions were all reproduced during the test for min(m,n,k) = 2, where am, bn, and ck were considered up to 271.

References

  1. ↑ 1.0 1.1 Adam S. Sikora (Oct 2024). Fermat-Catalan and Tijdeman-Zagier Conjectures for Products (Technical Report). 
  2. ↑ Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre, The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 978-0-691-11880-2 .
  3. ↑ Frits Beukers (Jan 1998). "The Diophantine equation Axp+Byq=Czr". Duke Math. J. 91 (1): 61–88. doi:10.1215/S0012-7094-98-09105-0.  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."
  4. ↑ Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society 27 (6): 513–43. doi:10.1112/blms/27.6.513. 
  5. ↑ Elkies, Noam D. (2007). "The ABC's of Number Theory". The Harvard College Mathematics Review 1 (1). http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2. 
  6. ↑ "Lecture on the abc conjecture and some of its consequences". Mathematics in the 21st century. Springer Proc. Math. Stat.. 98. Basel: Springer. 2015. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3. http://www.imj-prg.fr/~michel.waldschmidt/articles/pdf/abcLahoreProceedings.pdf. 
  7. ↑ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (Aug 2005). Twists of X(7) and primitive solutions to x2 + y3 = z7 (Report).  Here: p.3, table 1.
  8. ↑ 8.0 8.1 Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (2007). "Twists of X(7) and primitive solutions to x2 + y3 = z7". Duke Math. J. 137: 103–158. 
  9. ↑ 9.0 9.1 Nils Bruin (1999). "The Diophantine Equations x2 ± y4 = ± z6 and x2 + y8 = z3". Compositio Mathematica 118: 305–321. https://www.cambridge.org/core/journals/compositio-mathematica/article/diophantine-equations-x2-y4z6-and-x2y8-z3/A1F11FEADBCFE720756819531A5E9CAE. 
  10. ↑ 10.0 10.1 Nils Bruin (2003). "Chabauty methods using elliptic curves". J. Reine Angew. Math. 562: 27–49. https://www.cecm.sfu.ca/~nbruin/bruin_crelle.pdf. 
  11. ↑ Nils Bruin (2004). "Visualising Sha[2 in abelian surfaces"]. Math. Comp. (electronic) 73: 1459–1476. https://www.ams.org/journals/mcom/2004-73-247/S0025-5718-04-01633-3/S0025-5718-04-01633-3.pdf. 
  12. ↑ Imin Chen (2007). "On the equation s2 + y2p = α3". Math. Comp. 77 (262): 1223–1227. 
  13. ↑ Jordan S. Ellenberg (2004). "Galois representations attached to Q-curves and the generalized Fermat equation A4 + B2 = Cp". Amer. J. Math. 126: 763–787. https://people.math.wisc.edu/~ellenberg/A4B2Cp.pdf. 
  14. ↑ "follows easily from" Michael A. Bennett and Chris M. Skinner (2004). "Ternary Diophantine equations via Galois representations and modular forms". Canad. J. Math. 56 (1): 23–54. https://personal.math.ubc.ca/~bennett/BS.pdf. 
  15. ↑ the special case (4,n,4) was done earlier in Henri Darmon (1993). "The equation x4 - y4 = zp". C. R. Math. Rep. Acad. Sci. Canada 15: 286–290. https://www.math.mcgill.ca/darmon/pub/Articles/Research/10.Fermat-44p/paper.pdf. 
  16. ↑ 16.0 16.1 Henri Darmon and Loc Merel (1997). "Winding quotients and some variants of Fermats last theorem". J. Reine Angew. Math. 490: 81–100. https://eudml.org/doc/153941. 
  17. ↑ 17.0 17.1 Nils Bruin (2000). "On powers as sums of two cubes". in Wieb Bosma. Algorithmic Number Theory – 4th Intnl. Symp. ANTS. LNCS. 1838. Leiden: Springer. pp. 169–184. https://www.cecm.sfu.ca/~nbruin/eq33p.ps.gz. 
  18. ↑ Alain Kraus (1998). "Sur l'équation a3 + b3 = cp". Experiment. Math. 7: 1–13. https://projecteuclid.org/journals/experimental-mathematics/volume-7/issue-1/Sur-l%C3%A9quation-asp-3bsp-3csp-p/em/1047674269.pdf. 
  19. ↑ Michael A. Bennett (2006). "The equation x2n + y2n = z5". Journal de Théorie des Nombres de Bordeaux 18: 315–321. https://personal.math.ubc.ca/~bennett/bennetjtnb.pdf. 
  20. ↑ Andrew Wiles (1995). "Modular Elliptic Curves and Fermat's Last Theorem". Annals of Mathematics 142: 443–551. 
  21. ↑ Richard Taylor and Andrew Wiles (1995). "Ring-theoretic properties of certain Hecke algebras". Ann. of Math. (2) 141: 553–572. https://staff.fnwi.uva.nl/a.l.kret/Galoistheorie/taylor-wiles.pdf. 



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