Fermat-Catalan conjecture

The Fermat-Catalan conjecture is an open conjecture of number theory .

It gets its name from the fact that its formulation includes ideas from the Fermat conjecture and the Catalan conjecture . The conjecture says that there are only a finite number of solutions with ${\displaystyle a,b,c,m,n,k\in \mathbb {N} }$

${\displaystyle a^{m}+b^{n}=c^{k}}$,

where are coprimary to each other and ${\displaystyle a,b,c}$

${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}<1}$.

The latter condition excludes the Pythagorean triples ( with infinitely many solutions) and some other cases with infinitely many solutions. The inequality is exactly fulfilled by and permutations, each with an infinite number of solutions, and by (with permutations), each with a finite number of solutions. ${\displaystyle m=n=k=2}$ ${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}>1}$ ${\displaystyle (m,n,k)=(2,2,k),(2,3,3),(2,3,4),(2,3,5)}$ ${\displaystyle {\frac {1}{m}}+{\frac {1}{n}}+{\frac {1}{k}}=1}$ ${\displaystyle (m,n,k)=(2,4,4),(2,3,6),(3,3,3)}$

The case of the Catalan's conjecture, which has now been proven, is where one of the is the same . The only solution is by guessing ${\displaystyle a,b,c}$ ${\displaystyle 1}$

${\displaystyle 1^{m}+2^{3}=3^{2}\;}$.

Strictly speaking, infinitely many provide a solution, but this is also ruled out as a trivial special case. ${\displaystyle m>6}$

The other known cases are (as of 2015):

${\displaystyle 2^{5}+7^{2}=3^{4}\;}$

${\displaystyle 13^{2}+7^{3}=2^{9}\;}$

${\displaystyle 2^{7}+17^{3}=71^{2}\;}$

${\displaystyle 3^{5}+11^{4}=122^{2}\;}$

${\displaystyle 33^{8}+1549034^{2}=15613^{3}\;}$

${\displaystyle 1414^{3}+2213459^{2}=65^{7}\;}$

${\displaystyle 9262^{3}[phone]^{2}=113^{7}\;}$

${\displaystyle 17^{7}+76271^{3}=21063928^{2}\;}$

${\displaystyle 43^{8}+96222^{3}=30042907^{2}\;}$

The last and largest five solutions on the list are from Frits Beukers and Don Zagier .

According to a theorem by Henri Darmon and Andrew Granville based on Faltings' theorem, there are only finitely many solutions to fixed ones. The Fermat-Catalan conjecture also asserts finitude for an infinite number of possible exponents. ${\displaystyle n,m,k}$

The Fermat-Catalan conjecture follows from the abc conjecture .

According to Andrew Beal's conjecture , one of the exponents must be in the Fermat-Catalan conjecture . ${\displaystyle 2}$

Special values ​​of the exponents

The finiteness of the solutions has been examined for specific combinations of conjecture exponents , including: ${\displaystyle (m,n,k)}$

• (2, 3, 7) by Bjorn Poonen et al. a. (2005)
• The Fermat conjecture, proved by Andrew Wiles , treats the case (k, k, k) with no solution for ${\displaystyle k\geq 3}$
• Henri Darmon and Loïc Merel treated the case (k, k, 2) and (k, k, 3) and showed that there are no solutions for (k, k, 3), and for (k, k, 2) for . ${\displaystyle k\geq 3}$ ${\displaystyle k\geq 4}$
• (2n, 2n, 5) by Michael Bennett
• (2,4, n) by Jordan S. Ellenberg , Ellenberg / Bennett / Ng and Bruin and (2, n, 4) by Bennett and Bennett / Skinner
• (2,6, n) from Bennett / Chen and Bruin
• (2, n, 6), (3,3,2n), (3,6, n), (2, 2n, k) for k = 9, 10 or 15, (4, 2n, 3) from Bennett, I. Chen, S. Dahmen, S. Yazdani
• (2,4,7) from Ghioca
• (2,3,8), (2,3,9), (2,4,5), (2,4,6), (3,3,4), (3,3,5) from Bruin ( 2004)
• (5,5,7), (7,7,5) from Dahmen / Siksek
• (3,4,5) Siksek / Stoll

Henri Darmon (2012) pursues a program of generalizing the Frey curve of the Fermat problem (to Frey-Abel varieties) in order to investigate the generalized Fermat equations . ${\displaystyle (p,p,r)}$ ${\displaystyle r>3}$

Web links

• Eric W. Weisstein : Fermat-Catalan Conjecture . In: MathWorld (English).

Individual evidence

1. ↑ ^{a b} Beukers: The ABC conjecture . (PDF) Presentation slides 2005
2. ↑ The ten cases were listed as early as 1995 by Darmon, Granville. In: Bulletin of the London Mathematical Society , Volume 27, 1995, pp. 513-543. You can also find them in Beukers' lecture on the ABC Hypothesis 2005.
3. ^ R. Daniel Mauldin: A generalization of Fermat's problem: The Beal conjecture and prize problem . In: Notices AMS , December 1997, No. 11, p. 1437, ams.org (PDF)
4. ↑ Darmon, Granville: On the equations zm = F (x, y) and Axp + Byq = Czr . In: Bulletin of the London Mathematical Society , Volume 27, 1995, pp. 513-543
5. ^ Poonen, Edward Schaefer, Michael Stoll: Twists of X (7) and primitive solutions of ${\displaystyle x^{2}+y^{3}=z^{7}}$ . arxiv : math / 0508174
6. ↑ Darmon, Merel: Winding Quotients and Some Variants of Fermat's Last Theorem . In: J. reine angew. Math. , Volume 490, 1997, pp. 81-100, SUB Göttingen
7. ↑ Michael Bennett, with Chen, Dahmen, Yazdani: The Generalized Fermat equation: a progress report . (PDF) Hawaii-Manoa 2012, presentation slides
8. ^ N. Bruin: Visualizing Sha [2] in Abelian Surfaces . In: Math. Comput. , Volume 73, 2004, pp. 1459-1476