Catalan's conjecture (or MihÄilescu's theorem) is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
that was
conjectured
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
by the mathematician
EugĂšne Charles Catalan
EugĂšne Charles Catalan (30 May 1814 â 14 February 1894) was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic ...
in 1844 and proven in 2002 by
Preda MihÄilescu
Preda V. MihÄilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture.
Biography
Born in Bucharest,Stewart 2013 he is the brother of VintilÄ MihÄilescu.
After leaving Romania in ...
at
Paderborn University
Paderborn University (german: UniversitÀt Paderborn) is one of the fourteen public research universities in the state of North Rhine-Westphalia in Germany. It was founded in 1972 and 20,308 students were enrolled at the university in the winter ...
. The integers 2
3 and 3
2 are two
perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...
s (that is, powers of exponent higher than one) of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s whose values (8 and 9, respectively) are consecutive. The theorem states that this is the ''only'' case of two consecutive perfect powers. That is to say, that
History
The history of the problem dates back at least to
Gersonides
Levi ben Gershon (1288 â 20 April 1344), better known by his Graecized name as Gersonides, or by his Latinized name Magister Leo Hebraeus, or in Hebrew by the abbreviation of first letters as ''RaLBaG'', was a medieval French Jewish philosoph ...
, who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when
Victor-Amédée Lebesgue
Victor-AmĂ©dĂ©e Lebesgue, sometimes written Le Besgue, (2 October 1791, Grandvilliers (Oise) â 10 June 1875, Bordeaux ( Gironde)) was a mathematician working on number theory. He was elected a member of the AcadĂ©mie des sciences in 1847.
Se ...
dealt with the case ''b'' = 2.
In 1976,
Robert Tijdeman
Robert Tijdeman (born 30 July 1943 in Oostzaan, North Holland) 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 c ...
applied
Baker's method
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendent ...
in
transcendence theory to establish a bound on a,b and used existing results bounding ''x'',''y'' in terms of ''a'', ''b'' to give an effective upper bound for ''x'',''y'',''a'',''b''. Michel Langevin computed a value of
for the bound, resolving Catalan's conjecture for all but a finite number of cases.
Catalan's conjecture was proven by
Preda MihÄilescu
Preda V. MihÄilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture.
Biography
Born in Bucharest,Stewart 2013 he is the brother of VintilÄ MihÄilescu.
After leaving Romania in ...
in April 2002. The proof was published in the ''
Journal fĂŒr die reine und angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal fĂŒr die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
'', 2004. It makes extensive use of the theory of
cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers.
Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
s and
Galois module
In mathematics, a Galois module is a ''G''-module, with ''G'' being 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 i ...
s. An exposition of the proof was given by
Yuri Bilu Yuri may refer to:
People and fictional characters
Given name
*Yuri (Slavic name), the Slavic masculine form of the given name George, including a list of people with the given name Yuri, Yury, etc.
*Yuri (Japanese name), also YĆ«ri, feminine Jap ...
in the
SĂ©minaire Bourbaki. In 2005, MihÄilescu published a simplified proof.
Pillai's conjecture
Pillai's conjecture concerns a general difference of perfect powers : it is an open problem initially proposed by
S. S. Pillai, who conjectured that the gaps in the sequence of perfect powers tend to infinity. This is equivalent to saying that each positive integer occurs only finitely many times as a difference of perfect powers: more generally, in 1931 Pillai conjectured that for fixed positive integers ''A'', ''B'', ''C'' the equation
has only finitely many solutions (''x'', ''y'', ''m'', ''n'') with (''m'', ''n'') â (2, 2). Pillai proved that the difference
for any λ less than 1, uniformly in ''m'' and ''n''.
The general conjecture would follow from the
ABC conjecture
The ''abc'' conjecture (also known as the OesterlĂ©âMasser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph OesterlĂ© and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b' ...
.
Paul ErdĆs
Paul ErdĆs ( hu, ErdĆs PĂĄl ; 26 March 1913 â 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
conjectured that the ascending sequence of perfect powers satisfies for some positive constant ''c'' and all sufficiently large ''n''.
Pillai's conjecture means that for every natural number ''n'', there are only finitely many pairs of perfect powers
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer Exponentiation, power of another integer greater than one. More ...
with difference ''n''. The list below shows, for ''n'' †64, all solutions for perfect powers less than 1018, as . See also for the smallest solution (> 0).
See also
*Beal's conjecture
The Beal conjecture is the following conjecture in number theory:
:If
:: A^x +B^y = C^z,
:where ''A'', ''B'', ''C'', ''x'', ''y'', and ''z'' are positive integers with ''x'', ''y'', ''z'' â„ 3, then ''A'', ''B'', and ''C'' have a common prime ...
* Equation xy = yx
*FermatâCatalan conjecture
In number theory, the FermatâCatalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation
has only finitely many solutions (''a'',''b'',''c'',''m'',''n'',' ...
*Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form ''y''2 = ''x''3 + ''n'', where ''n'' is a fixed non-zero integer.
These curves were closely studied by Louis Mordell, from the point of view of determining their integer points. He sho ...
*RamanujanâNagell equation In mathematics, in the field of number theory, the RamanujanâNagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be so ...
*StĂžrmer's theorem
In number theory, StĂžrmer's theorem, named after Carl StĂžrmer, 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 equ ...
*Tijdeman's theorem
In 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 equation
:y^m = x^n + 1,
for ...
Notes
References
*
*
*
*
*
*
* Predates MihÄilescu's proof.
*
External links
*
Ivars Peterson's MathTrek
On difference of perfect powers
* Jeanine Daems
A Cyclotomic Proof of Catalan's Conjecture
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Conjectures
Conjectures that have been proved
Diophantine equations
Theorems in number theory