Catalan's conjecture states that 8 and 9 are the only consecutive integer powers. More formally, it claims that the only solution to x^p-y^q=1 (where x, y, p and q are integers greater than 1) is 3²-2³=1.

Whilst most number theorists believe Catalan's conjecture to be true, as of 2001 no-one has yet found a proof. Thanks to Robert Tijdeman, it has been known since 1976 that the equation above has only a finite number of solutions. Preda Mihailescu has also proven that if a second solution to the equation exists, then p and q must be double Wieferich primes. This result, in combination with some bounds on p and q proven by Maurice Mignotte, has brought the hunt for counterexamples to Catalan's conjecture within the capabilities of distributed brute force searching. (See http://catalan.ensor.org if you want to contribute computing time to this effort.) However, the amount of computation required is still huge, and it may turn out that a theoretical advance resolves Catalan's conjecture before an exhaustive computer search is complete. (In fact, Mihailescu claimed in May 2002 to have proved the conjecture; the proof is currently being verified.)

Catalan’s Conjecture

Mihailescu’s paper has now been verified as correct and proof of Catalan’s conjecture (Mihailescu 2004; Metsänkylä 2003).

This manuscript was reportedly sent on April 2002 to several mathematicians, (Poorten 2002) along with an expository paper / summary by Yuri Bilu (Bilu and Collectif 2004).

As usual in mathematics and science, the proof wasn’t created in a vacuum. There had been several papers on possible exceptions, special cases and conditions for the conjecture to hold (Weisstein 2002).

References and Bibliography

Bilu, Yuri, and Collectif. 2004. “Catalan’s Conjecture.” In Séminaire Bourbaki : Volume 2002/2003, Exposés 909-923, 1–26. Astérisque 294. Paris: Association des amis de Nicolas Bourbaki, Société mathématique de France. http://www.numdam.org/item/SB_2002-2003__45__1_0.

Metsänkylä, Tauno. 2003. “Catalan’s Conjecture: Another Old Diophantine Problem Solved.” Bulletin of the American Mathematical Society 41 (01): 43–58. https://doi.org/10.1090/s0273-0979-03-00993-5.

Mihailescu, P. 2004. “Primary Cyclotomic Units and a Proof of Catalans Conjecture.” Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal) 2004 (572). https://doi.org/10.1515/crll.2004.048.

Poorten, Alf van der. 2002. “Concerning: Catalan’s Conjecture Proved?” May 5, 2002. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;1a7d7d4.0205&S=.

Weisstein, Eric W. 2002. “MathWorld Headline News—Draft Proof of Catalan’s Conjecture Circulated.” May 5, 2002. https://mathworld.wolfram.com/news/2002-05-05/catalan/.

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