# Catalan's conjecture

Catalan's conjecture is the hypothesis, put forward by the Belgian mathematician Eugène Catalan
(1814–1894) in 1844, that 8 (= 2^{3}) and 9 (= 3^{2})
are the only pair of consecutive powers. In other words, the Catalan equation
for prime numbers *p* and *q* and positive integers *x* and *y*:

*x ^{ p}* -

*y*= 1

^{ q}

has only the one solution:

3^{2} - 2^{3} = 1.

In 1976 R. Tijdeman took the first major step toward showing this is true
by proving that for any solution, *y ^{ q}* is less than

*e*to the power

*e*to the power

*e*to the power

*e*to the power 730 (a huge number!). Since then this bound has been reduced many times, and it is now know that the larger of

*p*and

*q*is at most 7.78 × 10

^{16}and the smaller is at least 10

^{7}. On April 18, 2002, the Romanian number theorist Preda Mihailescu sent a manuscript to several mathematicians with a proof of the entire conjecture together with an analysis by Yuri Bilu. This proof was verified other mathematicians and published in 2004.

Solutions to Catalan's conjecture and Fermat's
last theorem are special cases of the **Fermat-Catalan equation**:

*x ^{ p}* +

*y*=

^{q}*z*

^{r}

Where *x*, *y*, and *z* are positive, coprime integers and the exponents are all primes with

1/*p* + 1/*q* + 1/*r* < 1.

The **Fermat-Catalan conjecture** is that there are only finitely
many solutions to this system. These solutions include: 1^{p} + 2^{3} = 3^{2} (*p* ≥ 2); 2^{5} + 7^{2} = 3^{4}; 13^{2} + 7^{3} = 2^{9}; 2^{7} + 7^{3} = 71^{2}; 3^{5} + 17^{3} = 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}; and 43^{8} + 96222^{3} = 30042907^{2}.

### Reference

1. Ribenboim, P. *Catalan's Conjecture*. Boston, Mass.: Academic
Press, 1994.