# Catalan's problem

In 1844, the Belgian mathematician Eugène Charles Catalan conjectured that 8 and 9 are the only pair of consecutive powers. That is, the **Catalan equation** for primes *p* and *q* and positive integers *x* and *y*:

x^{p}-y^{q}= 1

has only the one solution:

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

In 1976 R. Tijdeman took the first major step toward showing this by proving that for any solution
*y*^{q} is less than e^e^e^e^730 (a huge number!) [Guy94].
Since then this bound has been reduced many times, and we now know that the larger of *p* and *q*
is at most 7.78^{.}10^{16} and the
smaller is at least 10^{7} [Mignotte2000].

On April 18, 2002, Preda Mihailescu published his completed proof [Mihailescu2003] which begins by showing that about solutions to this problem is that any solutions other than the pair
(*p*,*q*) = (2,3) must satisfy both of:

p^{q-1}≡ 1 (modq^{2})

q^{p-1}≡ 1 (modp^{2})

That is, *p* is a Wieferich prime base *q*, and *q*
is a Wieferich prime base *p*. (Others had shown this for certain subsets of the primes.)

#### Fermat-Catalan equation

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*, *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}+ 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}

**See Also:** FermatsLastTheorem, BealsConjecture, WieferichPrime

**References:**

- Catalan1844
E. Catalan, "Note extraite d'une lettre adressée à l'édite,"J. reine angew. Mathematik,27(1844) 192.- CP2001 (p279-381)
R. CrandallandC. Pomerance,Prime numbers: a computational perspective, Springer-Verlag, 2001. New York, NY, pp. xvi+545, ISBN 0-387-94777-9.MR 2002a:11007(Abstract available) [This is a valuable text written by true experts in two different areas: computational and theoretical respectively. There is now a second edition [CP2005].]- Guy94 (section D9)
R. K. Guy,Unsolved problems in number theory, Springer-Verlag, 1994. New York, NY, ISBN 0-387-94289-0.MR 96e:11002[An excellent resource! Guy briefly describes many open questions, then provides numerous references. See his newer editions of this text.]- Mignotte2000
M. Mignotte,Catalan's equation just before 2001. In "Number theory (Turku, 1999)," M. Jutila and T. Metsänkylä editors, de Gruyter, 2001. Berlin,MR 2002g:11034- Mihailescu2003
P. Mihailescu, "A class number free criterion for Catalan's conjecture,"J. Number Theory,99:2 (2003) 225--231.MR 1 968 450- Peterson2000
I. Peterson, "Prime proof zeros in on crucial numbers,"Science News,158(December 2000) 357. Short note that Miailescu showed solutions to Catalan's are Wierferich double primes.- Ribenboim1994
P. Ribenboim,Catalan's conjecture: are 8 and 9 the only consecutive powers?, Academic Press, Boston, MA, 1994. pp. xvi+364, ISBN 0-12-587170-8.MR 95a:11029