Fermat–Catalan 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

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

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. .

Known solutions

As of 2015 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:[1][2]

(for to satisfy Eq. 2)

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.[3][4]: 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.[5]

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.[6][7] list exponent triples where the solutions have been determined:[note 1]   {2,3,7},[7]   {2,3,8},[8][9]   {2,3,9},[10]   {2,2q,3} for prime 7<q<1000 with q≠31,[11]   {2,4,5},[9]   {2,4,6},[8]   (2,4,7),   (2,4,q) for prime q≥211,[12]   (2,n,4),[13][14]   {2,n,n},[15]   {3,3,4},[16]   {3,3,5},[16]   {3,3,q} for 17≤q≤10000,[17]   {3,n,n},[15]   {2n,2n,5},[18]   {n,n,n}.[19][20]   For each of these exponent triples, if there is some solution at all, it is listed among those in section § Known solutions.

See also

Notes

  1. ^ The notation "{p,q,r}" means that the solutions have been determined for every permutation of (p,q,r).

References

  1. ^ Pomerance, Carl (2008), "Computational Number Theory", in Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), The Princeton Companion to Mathematics, Princeton University Press, pp. 361–362, ISBN 978-0-691-11880-2.
  2. ^ Frits Beukers (January 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."
  3. ^ 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.
  4. ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
  5. ^ Waldschmidt, Michel (2015). "Lecture on the conjecture and some of its consequences". Mathematics in the 21st century (PDF). Springer Proc. Math. Stat. Vol. 98. Basel: Springer. pp. 211–230. doi:10.1007/978-3-0348-0859-0_13. ISBN 978-3-0348-0858-3. MR 3298238.
  6. ^ Poonen, Bjorn; Schaefer, Edward F.; Stoll, Michael (August 2005). Twists of X(7) and primitive solutions to x2 + y3 = z7 (Report). arXiv:math/0508174. Here: p.3, table 1.
  7. ^ a b 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.
  8. ^ a b Nils Bruin (1999). "The Diophantine Equations x2 ± y4 = ± z6 and x2 + y8 = z3". Compositio Mathematica. 118: 305–321. MR 1711307.
  9. ^ a b Nils Bruin (2003). "Chabauty methods using elliptic curves" (PDF). J. Reine Angew. Math. 562: 27–49. MR 2011330.
  10. ^ Nils Bruin (2004). "Visualising Sha[2] in abelian surfaces" (PDF). Math. Comp. (electronic). 73: 1459–1476. MR 2047096.
  11. ^ Imin Chen (2007). "On the equation s2 + y2p = α3". Math. Comp. 77 (262): 1223–1227.
  12. ^ Jordan S. Ellenberg (2004). "Galois representations attached to Q-curves and the generalized Fermat equation A4 + B2 = Cp" (PDF). Amer. J. Math. 126: 763–787. JSTOR 40067960. MR 2075481.
  13. ^ "follows easily from" Michael A. Bennett and Chris M. Skinner (2004). "Ternary Diophantine equations via Galois representations and modular forms" (PDF). Canad. J. Math. 56 (1): 23–54. MR 2031121.
  14. ^ the special case (4,n,4) was done earlier in Henri Darmon (1993). "The equation x4 - y4 = zp" (PDF). C. R. Math. Rep. Acad. Sci. Canada. 15: 286–290. MR 1260076.
  15. ^ a b Henri Darmon and Loc Merel (1997). "Winding quotients and some variants of Fermats last theorem". J. Reine Angew. Math. 490: 81–100. MR 1468926.
  16. ^ a b Nils Bruin (2000). "On powers as sums of two cubes". In Wieb Bosma (ed.). Algorithmic Number Theory – 4th Intnl. Symp. ANTS. LNCS. Vol. 1838. Leiden: Springer. pp. 169–184. MR 1850605.
  17. ^ Alain Kraus (1998). "Sur l'équation a3 + b3 = cp" (PDF). Experiment. Math. 7: 1–13. MR 1618290.
  18. ^ Michael A. Bennett (2006). "The equation x2n + y2n = z5" (PDF). Journal de Théorie des Nombres de Bordeaux. 18: 315–321.
  19. ^ Andrew Wiles (1995). "Modular Elliptic Curves and Fermat's Last Theorem". Annals of Mathematics. 142: 443–551.
  20. ^ Richard Taylor and Andrew Wiles (1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Ann. of Math. (2). 141: 553–572. MR 1333036.
This article is issued from Wikipedia. The text is available under Creative Commons Attribution-Share Alike 4.0 unless otherwise noted. Additional terms may apply for the media files.