number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

, 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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...

and of

Catalan's conjecture Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ...

, hence the name. The conjecture states that the equation 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 mathematics, 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 equivale ...

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'') or with ''m'', ''n'', and ''k'' all equal to two (for the infinitely many known

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

s).

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 conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are ...

, 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 in ...

. 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 In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and ...

, 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 The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' ...

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

Known solutions

Partial results

See also

References

External links