
Glossary: Prime Pages: Top 5000: 
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} = 1has only the one solution: 3^{2}  2^{3} = 1In 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^{q1} 1 (mod q^{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 ofthe primes.) FermatCatalan equationSolutions to Catalan's conjecture and Fermat's Last Theorem are special cases of the FermatCatalan 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 FermatCatalan conjecture is that there are only finitely many solutions to this system. These solutions include: 1^{p} + 2^{3} = 3^{2} (p > 2)
See Also: FermatsLastTheorem, BealsConjecture, WieferichPrime References:
Chris K. Caldwell © 19992016 (all rights reserved)
