number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, the Fermat–Catalan conjecture is a generalization of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

and of Catalan's conjecture. The conjecture states that the

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

has only finitely many solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with distinct triplets of values (''a''^''m'', ''b''^''n'', ''c''^''k'') where ''a'', ''b'', ''c'' are positive

coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

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'' = ''a''^''m'' + ''b''^''n''), with ''m''=''n''=''k''=2 (for the infinitely many

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...

s), and e.g.

7^5 + 393^3 = 7792^2

Known solutions

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

1^m+2^3=3^2\;

(for

m>6

to satisfy Eq. 2) :

2^5+7^2=3^4\;

7^3+13^2=2^9\;

2^7+17^3=71^2\;

3^5+11^4=122^2\;

33^8+1549034^2=15613^3\;

1414^3+2213459^2=65^7\;

9262^3+15312283^2=113^7\;

17^7+76271^3=21063928^2\;

43^8+96222^3=30042907^2\;

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 Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was Conjecture, conjectured by the mathematician Eugène Charles Catalan in 1844 and mathematical proof, proven in 2002 by Preda Mihăilescu at Paderborn Univers ...

, proven in 2002 by

Preda Mihăilescu Preda V. Mihăilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture. Biography Born in Bucharest,Stewart 2013 he is the brother of Vintilă Mihăilescu. After leaving Romania i ...

. 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'', ''b''^''n'', ''c''^''k'').

Partial results

It is known by the Darmon–Granville theorem, which uses

Faltings's theorem Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field \mathbb of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and know ...

, 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. However, the full Fermat–Catalan conjecture is stronger as it allows for the exponents ''m'', ''n'' and ''k'' to vary. The

abc conjecture ABC are the first three letters of the Latin script. ABC or abc may also refer to: Arts, entertainment and media Broadcasting * Aliw Broadcasting Corporation, Philippine broadcast company * American Broadcasting Company, a commercial American ...

implies the Fermat–Catalan conjecture. 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.

References

External links

''Perfect Powers: Pillai's works and their developments''. Waldschmidt, M.
* {{DEFAULTSORT:Fermat-Catalan conjecture Conjectures Unsolved problems in number theory Diophantine equations Abc conjecture

Known solutions

Partial results

See also

References

External links