Fermat–Catalan conjecture: Difference between revisions
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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. |
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. |
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== Fermat–Catalan–LPS conjecture == |
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The '''Fermat–Catalan–LPS conjecture''' is a generalization of Fermat–Catalan conjecture and of [[Lander, Parkin, and Selfridge conjecture]], this conjecture is that the equation |
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<math>\sum_{i=1}^m {a_i}^{b_i} = \sum_{j=1}^n {c_j}^{d_j}</math> |
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has only finitely many solutions with ''m'', ''n'', ''a<sub>i</sub>'', ''b<sub>i</sub>'', ''c<sub>j</sub>'', ''d<sub>j</sub>'' all positive integers, <math>\gcd(a_1,a_2,\dots,a_m,c_1,c_2,\dots,c_n) = 1</math>, <math>\sum A \neq \sum C</math> for all [[nonempty]] [[proper subset]] ''A'' of {''a<sub>i</sub><sup>b<sub>i</sub></sup>'', ''i'' = 1, 2, ..., ''m''} and all nonempty proper subset ''C'' of {''c<sub>j</sub><sup>d<sub>j</sub></sup>'', ''j'' = 1, 2, ..., ''n''}, <math>\sum_{i=1}^m \frac{1}{b_i} + \sum_{j=1}^n \frac{1}{d_j} < 1</math> |
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The Fermat–Catalan conjecture is that this equation has only finitely many solutions with <math>m+n=3</math>. |
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The Lander, Parkin, and Selfridge conjecture is that this equation has no solution with <math>b_1=b_2=\dots=b_m</math> and <math>d_1=d_2=\dots=d_n</math>. |
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==See also== |
==See also== |
Revision as of 09:14, 25 January 2021
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's last theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation
(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
(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) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples).
Known solutions
As of 2015, the following ten solutions to equation (1) which meet the criteria of equation (2) are known:[1]
- (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.[2][3]: 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.[4]
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.
Fermat–Catalan–LPS conjecture
The Fermat–Catalan–LPS conjecture is a generalization of Fermat–Catalan conjecture and of Lander, Parkin, and Selfridge conjecture, this conjecture is that the equation
has only finitely many solutions with m, n, ai, bi, cj, dj all positive integers, , for all nonempty proper subset A of {aibi, i = 1, 2, ..., m} and all nonempty proper subset C of {cjdj, j = 1, 2, ..., n},
The Fermat–Catalan conjecture is that this equation has only finitely many solutions with .
The Lander, Parkin, and Selfridge conjecture is that this equation has no solution with and .
See also
- Sums of powers, a list of related conjectures and theorems
References
- ^ 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.
- ^ Darmon, H.; Granville, A. (1995). "On the equations zm = F(x, y) and Axp + Byq = Czr". Bulletin of the London Mathematical Society. 27: 513–43. doi:10.1112/blms/27.6.513.
- ^ Elkies, Noam D. (2007). "The ABC's of Number Theory" (PDF). The Harvard College Mathematics Review. 1 (1).
- ^ 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. MR 3298238.