Catalan's Conjecture
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Catalan's conjecture (or Mihăilescu'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 ...
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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
Eugène Charles Catalan Eugène Charles Catalan (30 May 1814 – 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 Mihăilescu Preda V. Mihăilescu (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 Vintilă Mihăilescu. After leaving Romania i ...
at
Paderborn University Paderborn University (german: Universität 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 23 and 32 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 ...
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 – 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 philosop ...
, 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 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 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 \exp \exp \exp \exp 730 \approx 10^ for the bound, resolving Catalan's conjecture for all but a finite number of cases. Catalan's conjecture was proven by
Preda Mihăilescu Preda V. Mihăilescu (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 Vintilă Mihăilescu. After leaving Romania i ...
in April 2002. The proof was published in the ''
Journal für die reine und angewandte Mathematik ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics''). History The journal wa ...
'', 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 o ...
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 ...
s. An exposition of the proof was given by Yuri Bilu in the Séminaire Bourbaki. In 2005, Mihăilescu 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 Subbayya Sivasankaranarayana Pillai (5 April 1901 – 31 August 1950) was an Indian mathematician specialising in number theory. His contribution to Waring's problem was described in 1950 by K. S. Chandrasekharan as "almost certainly his best p ...
, 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 Ax^n - By^m = C has only finitely many solutions (''x'', ''y'', ''m'', ''n'') with (''m'', ''n'') ≠ (2, 2). Pillai proved that the difference , Ax^n - By^m, \gg x^ 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 Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' ...
. Paul Erdős conjectured that the ascending sequence (a_n)_ of perfect powers satisfies a_ - a_n > n^c 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 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'' ≥ 3, then ''A'', ''B'', and ''C'' have a common prime ...
* Equation xy = yx *
Fermat–Catalan conjecture In number theory, the Fermat–Catalan 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 sh ...
*
Ramanujan–Nagell equation In mathematics, in the field of number theory, the Ramanujan–Nagell 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 s ...
*
Størmer's theorem In 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 e ...
*
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 Mihăilescu's proof. *


External links

*
Ivars Peterson's MathTrek

On difference of perfect powers
* Jeanine Daems
A Cyclotomic Proof of Catalan's Conjecture
{{Authority control Conjectures Conjectures that have been proved Diophantine equations Theorems in number theory