# Eugène Charles Catalan

Last updated
Eugène Charles Catalan
Born30 May 1814
Died 14 February 1894 (aged 79)
Nationality French, Belgian
Alma mater Ecole Polytechnique
Known for Catalan's conjecture, Catalan numbers
Scientific career
Fields Mathematics
Doctoral students François Deruyts
Charles Hermite
Constantin Le Paige

Eugène Charles Catalan (30 May 1814 – 14 February 1894) [1] was a French and Belgian mathematician who worked on continued fractions, descriptive geometry, number theory and combinatorics. His notable contributions included discovering a periodic minimal surface in the space ${\displaystyle \mathbb {R} ^{3}}$; stating the famous Catalan's conjecture, which was eventually proved in 2002; and, introducing the Catalan numbers to solve a combinatorial problem.

Belgium, officially the Kingdom of Belgium, is a country in Western Europe. It is bordered by the Netherlands to the north, Germany to the east, Luxembourg to the southeast, France to the southwest, and the North Sea to the northwest. It covers an area of 30,688 square kilometres (11,849 sq mi) and has a population of more than 11.4 million. The capital and largest city is Brussels; other major cities are Antwerp, Ghent, Charleroi and Liège.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction, the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction.

Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric problem solving. His first discoveries were in 1765 while he was working as a draftsman for military fortifications, although his findings were published later on.

## Biography

Catalan was born in Bruges (now in Belgium, then under Dutch rule even though the Kingdom of the Netherlands had not yet been formally instituted), 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 (1833). In December 1834 he was expelled along with most of the students in his year for political reasons; [2] he resumed his studies in January 1835, graduated that summer, and went on to teach at Châlons-sur-Marne. Catalan came back to the École Polytechnique, and, with the help of Liouville, obtained his degree in mathematics in 1841. He went on to Charlemagne College to teach descriptive geometry. Though he was politically active and strongly left-wing, leading him to participate in the 1848 Revolution, he had an animated career and also sat in the France's Chamber of Deputies. Later, in 1849, Catalan was visited at his home by the French Police, searching for illicit teaching material; however, none was found.

Bruges is the capital and largest city of the province of West Flanders in the Flemish Region of Belgium, in the northwest of the country.

The Kingdom of the Netherlands, commonly known as the Netherlands, is a sovereign state and constitutional monarchy with the large majority of its territory in Western Europe and with several small island territories in the Caribbean Sea, in the West Indies islands.

École polytechnique is a French public institution of higher education and research in Palaiseau, a suburb southwest of Paris. It is one of the most prestigious and selective French scientific and engineering schools, called grandes écoles in French. It is known for its ingénieur polytechnicien scientific degree program which is equivalent to both a bachelor and master of science. Its entrance exam, the X-ENS exam, is renowned for its selectivity with a little over 500 admitted students out of the 53 848 students enrolled in the preparatory programs for the French scientific and engineering schools entrance exams.

The University of Liège appointed him chair of analysis in 1865. In 1879, still in Belgium, he became journal editor where he published as a foot note Paul-Jean Busschop's theory after refusing it in 1873 - letting Busschop know that it was too empirical. In 1883, he worked for the Belgian Academy of Science in the field of number theory. He died in Liège, Belgium where he had received a chair.

The University of Liège (ULiège), in Liège, Wallonia, Belgium, is a major public university in the French Community of Belgium. Its official language is French. As of 2016, ULiège is ranked in the #251–300 category worldwide according to Times Higher Education, 265nd by QS World University Rankings, and between the 205th and 300th place by the Academic Ranking of World Universities. More than 2000 people, academics, scientists and technicians, are involved in research of a wide variety of subjects from basic research to applied research.

## Work

He worked on continued fractions, descriptive geometry, number theory and combinatorics. He gave his name to a unique surface (periodic minimal surface in the space ${\displaystyle \mathbb {R} ^{3}}$) that he discovered in 1855. Before that, he had stated the famous Catalan's conjecture, which was published in 1844 and was eventually proved in 2002, by the Romanian mathematician Preda Mihăilescu. He introduced the Catalan numbers to solve a combinatorial problem.

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.

Catalan's conjecture (or 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. The integers 23 and 32 are two powers of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers of

## Selected publications

• Théorèmes et Problèmes Géométrie élémentaire, Brussels, 2nd edition 1852, 6th edition 1879
• Éléments de géométrie, 1843, 2nd printing 1847
• Traité élémentaire de géométrie descriptive, 2 volumes 1850, 1852, 3rd edition 1867/1868, 5th edition 1881
• Nouveau manuel des aspirants au baccalauréat ès sciences, 1852 (12 editions published)
• Solutions des problèmes de mathématique et de physique donnés à la Sorbonne dans les compositions du baccalauréat ès sciences, 1855/56
• Manuel des candidats à l'École Polytechnique, 2 volumes, 1857-58
• Notions d'astronomie, 1860 (6 editions published)
• Traité élémentaire des séries, 1860
• Histoire d'un concours, 1865, 2nd edition 1867
• Cours d'analyse de l'université de Liège, 1870, 2nd edition 1880
• Intégrales eulériennes ou elliptiques, 1892

In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865.

Cassini's identity and Catalan's identity are mathematical identities for the Fibonacci numbers. The former is a special case of the latter, and states that for the nth Fibonacci number,

In mathematics, the Fibonacci numbers, commonly denoted Fn form a sequence, called the Fibonacci sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1. That is,

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