Mihailescu's theorem
Encyclopedia
Catalan's conjecture is a theorem
in number theory
that was conjectured by the mathematician Eugène Charles Catalan
in 1844 and proven in 2002 by Preda Mihăilescu
.
23 and 32 are two powers of natural number
s, 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
for x, a, y, b > 1 is x = 3, a = 2, y = 2, b = 3.
, who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3.
In 1976, Robert Tijdeman
applied methods from the theory of transcendental number
s 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 field
s and Galois module
s. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki.
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.
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
that was conjectured by the mathematician Eugène Charles Catalan
Eugène Charles Catalan
Eugène Charles Catalan was a French and Belgian mathematician.- Biography :Catalan was born in Bruges , the only child of a French jeweller by the name of Joseph Catalan, in 1814. In 1825, he traveled to Paris and learned mathematics at École Polytechnique, where he met Joseph Liouville...
in 1844 and proven in 2002 by Preda Mihăilescu
Preda Mihailescu
Preda V. Mihăilescu is a Romanian mathematician, best known for his proof of Catalan's conjecture.Born in Bucharest, he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving his Ph.D. from...
.
23 and 32 are two powers of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s, 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
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
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 GersonidesGersonides
Levi ben Gershon, better known by his Latinised name as Gersonides or the abbreviation of first letters as RaLBaG , philosopher, Talmudist, mathematician, astronomer/astrologer. He was born at Bagnols in Languedoc, France...
, who proved a special case of the conjecture in 1343 where x and y were restricted to be 2 or 3.
In 1976, Robert Tijdeman
Robert Tijdeman
Robert Tijdeman is a Dutch mathematician. Specializing in number theory, he is best known for his Tijdeman's theorem. He is a professor of mathematics at the Leiden University since 1975, and was chairman of the department of mathematics and computer science at Leiden from 1991 to 1993...
applied methods from the theory of transcendental number
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
s to show that there is an effectively computable constant C so that the exponents
Exponentiation
Exponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
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
Preda Mihailescu
Preda V. Mihăilescu is a Romanian mathematician, best known for his proof of Catalan's conjecture.Born in Bucharest, he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving his Ph.D. from...
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 field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s and Galois module
Galois module
In mathematics, a Galois module is a G-module where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module...
s. 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 each positive integer occurs only finitely many times as a difference of perfect powers. It is an open problem and is named for S. S. Pillai.Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
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
- Tijdeman's theoremTijdeman's theoremIn number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equationy^m = x^n + 1,\...
- Størmer's theoremStørmer's theoremIn number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations...
- Fermat–Catalan conjectureFermat–Catalan conjectureIn number theory, the Fermat–Catalan conjecture combines ideas of Fermat's last theorem and the Catalan conjecture, hence the name. The conjecture states that the equation...
- Beal's conjectureBeal's conjectureBeal's conjecture is a conjecture in number theory proposed by Andrew Beal in about 1993; a similar conjecture was suggested independently at about the same time by Andrew Granville....