Catalan's conjecture
Catalan's conjecture (occasionally now referred to as 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.
To understand the conjecture, notice that 23 and 32 are two powers of natural numbers, whose values 8 and 9 respectively are consecutive. The conjecture states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of
- xa − yb = 1
for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.
History
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3.
In 1974, Robert Tijdeman applied methods from the theory of transcendental numbers to show that there is an effectively computable constant C so that the exponents of all consecutive powers are less than C. As the results of a number of other mathematicians collectively had established a bound for the base dependent only on the exponents, this resolved Catalan's conjecture for all but a finite number of cases. However, the finite calculation required to complete the proof of the theorem was nonetheless too time-consuming to perform.
Catalan's conjecture was proved by Preda Mihăilescu in April 2002, so it is now sometimes called Mihăilescu's theorem. The proof was published in the Journal für die reine und angewandte Mathematik, 2004. It makes extensive use of the theory of cyclotomic fields and Galois modules. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki.
Pillai's conjecture
Pillai's conjecture concerns a general difference of perfect powers. It states that the differences in the sequence of all perfect powers tend to infinity, so that each given difference occurs only finitely many times. It is an open problem and is named for S. S. Pillai.
Paul Erdős conjectured that there is some positive constant c such that if d is the difference of a perfect power n, then d>nc for sufficiently large n.
See also
References
- Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine angew. Math. 572: 167–195. doi:10.1515/crll.2004.048.. MathSciNet Featured Review MR2076124.
- Paulo Ribenboim (1994). Catalan's Conjecture. Academic Press. ISBN 0-12-587170-8. Predates Mihăilescu's proof.
- Robert Tijdeman (1976). "On the equation of Catalan". Acta Arith. 29 (2): 197–209.
- Tauno Metsänkylä (2004). "Catalan's conjecture: another old Diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/S0273-0979-03-00993-5.
- Yuri Bilu (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque. 294: vii, 1–26.
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