Eugène Charles Catalan
Eugène Charles Catalan 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 R 3. 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 December 1834 he was expelled along with most of the students in his year for political reasons. Catalan came back to the École Polytechnique, 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 left-wing, leading him to participate in the 1848 Revolution, he had an animated career and sat in the France's Chamber of Deputies. In 1849, Catalan was visited at his home by the French Police, searching for illicit teaching material.
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 Belgium where he had received a chair. He worked on descriptive geometry, number theory and combinatorics, he gave his name to a unique surface that he discovered in 1855. Before that, he had stated the famous Catalan's conjecture, published in 1844 and was proved in 2002, by the Romanian mathematician Preda Mihăilescu, he introduced the Catalan numbers to solve a combinatorial problem. 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 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 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 Catalan solid Cassini and Catalan identities for Fibonacci numbers Catalan's constant Catalan number Catalan–Mersenne number/Catalan's Mersenne conjecture Catalan surface Catalan's conjecture Catalan's minimal surface O'Connor, John J..
Eugène Charles Catalan at the Mathematics Genealogy Project Catalan
Number theory
Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss 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. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
Wolfgang M. Schmidt
Wolfgang M. Schmidt is a mathematician working in the area of number theory, he studied mathematics at the University of Vienna, where he received his PhD, supervised by Edmund Hlawka, in 1955. Wolfgang Schmidt is professor at the University of Colorado at Boulder and a member of the Austrian Academy of Sciences and the Polish Academy of Sciences, he was awarded the eighth Frank Nelson Cole Prize in Number Theory for work on Diophantine approximation. He is known for his subspace theorem. In 1960, he proved that every normal number in base r is normal in base s if and only if log r / log s is a rational number, he proved the existence of T numbers. His series of papers on irregularities of distribution can be seen in J. Beck and W. Chen, Irregularities of Distribution, Cambridge University Press. Schmidt is in a small group of number theorists who have been invited to address the International Congress of Mathematicians three times; the others are Iwaniec and Tate. In 1986, Schmidt received the Humboldt Research Award and in 2003, he received the Austrian Decoration for Science and Art.
Schmidt holds honorary doctorates from the University of Ulm, the Sorbonne, the University of Waterloo, the University of Marburg and the University of York. In 2012 he became a fellow of the American Mathematical Society. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000 Equations Over Finite Fields: An Elementary Approach, 2nd edition, Kendrick Press 2004 Diophantine approximation: festschrift for Wolfgang Schmidt, Wolfgang M. Schmidt, H. P. Schlickewei, Robert F. Tichy, Klaus Schmidt, Springer, 2008, ISBN 978-3-211-74279-2
Abc conjecture
The abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser. It is stated in terms of three positive integers, a, b and c that are prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture states that d is not much smaller than c. In other words: if a and b are composed from large powers of primes c is not divisible by large powers of primes; the precise statement is given below. The abc conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves; the latter conjecture has more geometric structures involved in its statement in comparison with the abc conjecture. The abc conjecture and its versions express, in concentrated form, some fundamental feature of various problems in Diophantine geometry. A number of famous conjectures and theorems in number theory would follow from the abc conjecture or its versions. Goldfeld described the abc conjecture as "the most important unsolved problem in Diophantine analysis".
Various proofs of abc have been claimed but so far none is accepted by the mathematical community. Before we state the conjecture we introduce the notion of the radical of an integer: for a positive integer n, the radical of n, denoted rad, is the product of the distinct prime factors of n. For example rad = rad = 2, rad = 17, rad = rad = 2 · 3 = 6, rad = rad = 2 ⋅ 5 = 10. If a, b, c are coprime positive integers such that a + b = c, it turns out that "usually" c < rad. The abc conjecture deals with the exceptions, it states that: ABC Conjecture. For every positive real number ε, there exist only finitely many triples of coprime positive integers, with a + b = c, such that: c > rad 1 + ε. An equivalent formulation states that: ABC Conjecture II. For every positive real number ε, there exists a constant Kε such that for all triples of coprime positive integers, with a + b = c: c < K ε ⋅ rad 1 + ε. A third equivalent formulation of the conjecture involves the quality q of the triple, defined as q = log log .
For example, q = log / log = log / log = 0.46820... Q = log / log = log / log = 1.426565... A typical triple of coprime positive integers with a + b = c will have c < rad, i.e. q < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers. ABC Conjecture III. For every positive real number ε, there exist only finitely many triples of coprime positive integers with a + b = c such that q > 1 + ε. Whereas it is known that there are infinitely many triples of coprime positive integers with a + b = c such that q > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or q > 1.0001, etc. In particular, if the conjecture is true there must exist a triple which achieves the maximal possible quality q; the condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with rad < c. For example let: a = 1, b = 2 6 n − 1, c = 2 6 n, n > 1. First we note that b is divisible by 9: b = 2 6 n − 1 = 64 n − 1 = = 9 ⋅ 7 ⋅ Using this fact we calculate: rad = rad rad rad = rad rad rad ( 2 6 n
Perfect power
In mathematics, a perfect power is a positive integer that can be resolved into equal factors, whose root can be extracted. I.e. a positive integer that can be expressed as an integer power of another positive integer. More formally, n is a perfect power if there exist natural numbers m > 1, k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3 n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are considered perfect powers. A sequence of perfect powers can be generated by iterating through the possible values for k; the first few ascending perfect powers in numerical order are: 2 2 = 4, 2 3 = 8, 3 2 = 9, 2 4 = 16, 4 2 = 16, 5 2 = 25, 3 3 = 27, 2 5 = 32, 6 2 = 36, 7 2 = 49, 2 6 = 64, 4 3 = 64, 8 2 = 64, … The sum of the reciprocals of the perfect powers is 1: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = 1. Which can be proved as follows: ∑ m = 2 ∞ ∑ k = 2 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 ∑ k = 0 ∞ 1 m k = ∑ m = 2 ∞ 1 m 2 = ∑ m = 2 ∞ 1 m = ∑ m = 2 ∞ = 1.
The first perfect powers without duplicates are:, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024... The sum of the reciprocals of the perfect powers p without duplicates is: ∑ p 1 p = ∑ k = 2 ∞ μ ≈ 0.874464368 … where μ is the Möbius function and ζ is the Riemann zeta function. According to Euler, Goldbach showed that the sum of 1/p − 1 over the set of perfect powers p, excluding 1 and excluding duplicates, is 1: ∑ p 1 p − 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + ⋯ = 1; this is sometimes known as the Goldbach–Euler theorem. Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to k ≤ log 2 n. So if the divisors of n are n 1, n 2, …, n j one of the values n 1 2, n 2 2
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences cited as Sloane's, is an online database of integer sequences. It was maintained by Neil Sloane while a researcher at AT&T Labs. Foreseeing his retirement from AT&T Labs in 2012 and the need for an independent foundation, Sloane agreed to transfer the intellectual property and hosting of the OEIS to the OEIS Foundation in October 2009. Sloane is president of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional mathematicians and amateurs, is cited; as of September 2018 it contains over 300,000 sequences. Each entry contains the leading terms of the sequence, mathematical motivations, literature links, more, including the option to generate a graph or play a musical representation of the sequence; the database is searchable by subsequence. Neil Sloane started collecting integer sequences as a graduate student in 1965 to support his work in combinatorics; the database was at first stored on punched cards.
He published selections from the database in book form twice: A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and assigned M-numbers from M0000 to M5487; the Encyclopedia includes the references to the corresponding sequences in A Handbook of Integer Sequences as N-numbers from N0001 to N2372 The Encyclopedia includes the A-numbers that are used in the OEIS, whereas the Handbook did not. These books were well received and after the second publication, mathematicians supplied Sloane with a steady flow of new sequences; the collection became unmanageable in book form, when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, soon after as a web site. As a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998; the database continues to grow at a rate of some 10,000 entries a year.
Sloane has managed'his' sequences for 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100,000th sequence to the database, A100000, which counts the marks on the Ishango bone. In 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS.org was created to simplify the collaboration of the OEIS editors and contributors. The 200,000th sequence, A200000, was added to the database in November 2011. Besides integer sequences, the OEIS catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences: the sequence of numerators and the sequence of denominators. For example, the fifth-order Farey sequence, 1 5, 1 4, 1 3, 2 5, 1 2, 3 5, 2 3, 3 4, 4 5, is catalogued as the numerator sequence 1, 1, 1, 2, 1, 3, 2, 3, 4 and the denominator sequence 5, 4, 3, 5, 2, 5, 3, 4, 5.
Important irrational numbers such as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, it still uses a linear form of conventional mathematical notation. Greek letters are represented by their full names, e.g. mu for μ, phi for φ. Every sequence is identified by the letter A followed by six digits always referred to with leading zeros, e.g. A000315 rather than A315. Individual terms of sequences are separated by commas. Digit groups are not separated by periods, or spaces. In comments, etc. A represents the nth term of the sequence. Zero is used to represent non-existent sequence elements. For example, A104157 enumerates the "smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists." The value of a is 2. But there is no such 2×2 magic square, so a is 0; this special usage has a solid mathematical basis in certain counting functions.
For example, the totient valence function. There are 4 solutions for 4, but no solutions for 14, hence a of A014197 is 0—there are no solutions. −1 is used for this purpose instead, as in A094076. The OEIS ma
Paul Erdős
Paul Erdős was a renowned Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century, he was known both for his eccentric lifestyle. He devoted his waking hours to mathematics into his years—indeed, his death came only hours after he solved a geometry problem at a conference in Warsaw. Erdős pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, probability theory. Much of his work centered around discrete mathematics, cracking many unsolved problems in the field, he championed and contributed to Ramsey theory, which studied the conditions in which order appears. Overall, his work leaned towards solving open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed, he believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathematicians.
Erdős's prolific output with co-authors prompted the creation of the Erdős number, the number of steps in the shortest path between a mathematician and Erdős in terms of co-authorships. Paul Erdős was born in Budapest, Austria-Hungary, on 26 March 1913, he was the only surviving child of Lajos Erdős. His two sisters, aged 3 and 5, both died of scarlet fever a few days, his parents were both Jewish mathematics teachers. His fascination with mathematics developed early—he was left home by himself because his father was held captive in Siberia as an Austro-Hungarian POW during 1914–1920, causing his mother to have to work long hours to support their household, he taught himself to read through mathematics texts. By the age of four, given a person's age, he could calculate in his head how many seconds they had lived. Due to his sisters' deaths, he had a close relationship with his mother, with the two of them sharing the same bed until he left for college. Both of Erdős's parents were high school mathematics teachers, Erdős received much of his early education from them.
Erdős always remembered his parents with great affection. At 16, his father introduced him to two of his lifetime favorite subjects—infinite series and set theory. During high school, Erdős became an ardent solver of the problems proposed each month in KöMaL, the Mathematical and Physical Monthly for Secondary Schools. Erdős entered the University of Budapest at the age of 17. By the time he was 20, he had found a proof for Chebyshev's Theorem. In 1934, at the age of 21, he was awarded a doctorate in mathematics. Erdős's thesis advisor was Lipót Fejér, the thesis advisor for John von Neumann, George Pólya, Paul Turán; because he was a Jew, Erdős decided. Many members of Erdős' family, including two of his aunts, two of his uncles, his father, died in Budapest during the Holocaust, his mother survived in hiding. He was working at the Princeton Institute for Advanced Study at the time. Described by his biographer, Paul Hoffman, as "probably the most eccentric mathematician in the world," Erdős spent most of his adult life living out of a suitcase.
Except for some years in the 1950s, when he was not allowed to enter the United States based on the pretense that he was a Communist sympathizer, his life was a continuous series of going from one meeting or seminar to another. During his visits, Erdős expected his hosts to lodge him, feed him, do his laundry, along with anything else he needed, as well as arrange for him to get to his next destination. On 20 September 1996, at the age of 83, he had a heart attack and died while attending a conference in Warsaw; these circumstances were close to the way. He once said, I want to be giving a lecture, finishing up an important proof on the blackboard, when someone in the audience shouts out,'What about the general case?'. I'll turn to the audience and smile,'I'll leave that to the next generation,' and I'll keel over. Erdős never had no children, he is buried next to his father in grave 17A-6-29 at Kozma Utcai Temető in Budapest. For his epitaph, he suggested "I've stopped getting dumber.". His life was documented in the film N Is a Number: A Portrait of Paul Erdős, made while he was still alive, posthumously in the book The Man Who Loved Only Numbers.
Erdős' name contains the Hungarian letter "ő", but is incorrectly written as Erdos or Erdös either "by mistake or out of typographical necessity". Possessions meant little to Erdős. Awards and other earnings were donated to people in need and various worthy causes, he spent most of his life traveling between scientific conferences and the homes of colleagues all over the world. He earned enough in stipends from universities as a guest lecturer, from various mathematical awards, to fund his travels and basic needs, he would show up at a colleague's doorstep and announce "my brain is open", staying long enough to collaborate on a few papers before moving on a few days later. In many cases, he would ask the c