Fermat–Catalan conjecture#Known solutions

As of 2015 the following ten solutions to equation (1) which meet the criteria of equation (2) are known:{{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}}.{{cite journal | doi=10.1215/S0012-7094-98-09105-0 | author=Frits Beukers | author-link=Frits Beukers |title=The Diophantine equation Ax^p+By^q=Cz^r | 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."

:$1^m+2^3=3^2\backslash ;$ (for $m>6$ to satisfy Eq. 2)

:$2^5+7^2=3^4\backslash ;$

:$7^3+13^2=2^9\backslash ;$

:$2^7+17^3=71^2\backslash ;$

:$3^5+11^4=122^2\backslash ;$

:$33^8+1549034^2=15613^3\backslash ;$

:$1414^3+2213459^2=65^7\backslash ;$

:$9262^3+15312283^2=113^7\backslash ;$

:$17^7+76271^3=21063928^2\backslash ;$

:$43^8+96222^3=30042907^2\backslash ;$

The first of these (1^m + 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 (a^m, bⁿ, c^k).

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.{{cite journal |first1=H. |last1=Darmon |first2=A. |last2=Granville |title=On the equations z^m = F(x, y) and Ax^p + By^q = Cz^r |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 }}{{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}}{{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.{{cite book

| last = Waldschmidt | first = Michel | authorlink = Michel Waldschmidt

| contribution = Lecture on the $abc$ 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 }}

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.

• Sums of powers, a list of related conjectures and theorems

{{reflist}}

• [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}}

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Category:Conjectures

Category:Unsolved problems in number theory

Category:Diophantine equations

Category:Abc conjecture