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Guido Hoheisel
Guido Karl Heinrich Hoheisel (14 July 1894 – 11 October 1968) was a German mathematician and professor of mathematics at the University of Cologne. Academic life He did his PhD in 1920 from the University of Berlin under the supervision of Erhard Schmidt. During World War II Hoheisel was required to teach classes simultaneously at three universities, in Cologne, Bonn, and Münster. His doctoral students include Arnold Schönhage. Hoheisel contributed to the journal Deutsche Mathematik. Selected results Hoheisel is known for a result on gaps between prime numbers: He proved that if π(x) denotes the prime-counting function, then there exists a constant θ < 1 such that :π(''x'' + ''x''θ) − π(''x'') ~ ''x''θ/log(''x''), as ''x'' tends to infinity, implying that if ''p''''n'' denotes the ''n''-th then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Schönhage
Arnold Schönhage (born 1 December 1934 in Lockhausen, now Bad Salzuflen) is a German mathematician and computer scientist. Schönhage was professor at the Rheinische Friedrich-Wilhelms-Universität, Bonn, and also in Tübingen and Konstanz. He now lives near Bonn. Together with Volker Strassen he developed the Schönhage–Strassen algorithm for fast integer multiplication that has a run-time of '' O''(''N'' log ''N'' log log ''N''). Schönhage designed and implemented together with Andreas F. W. Grotefeld and Ekkehart Vetter a multitape Turing machine, called TP, in software. The machine is programmed in TPAL, an assembler language In computer programming, assembly language (or assembler language, or symbolic machine code), often referred to simply as Assembly and commonly abbreviated as ASM or asm, is any low-level programming language with a very strong correspondence be .... They implemented numerous numerical algorithms including the Sch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sufficiently Large
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of \mathbb). Notation The general form where the phrase eventually (or sufficiently large) is found appears as follows: :P is ''eventually'' true for x (P is true for ''sufficiently large'' x), where \forall and \exists are the universal and existential quantifiers, which is actually a shorthand for: :\exists a \in \mathbb such that P is true \forall x \ge a or somewhat more formally: :\exists a \in \mathbb: \forall x \in \mathbb:x \ge a \Rightarrow P(x) This does not necessarily mean that any particular value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately : \frac x where log is the natural logarithm, in the sense that :\lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is :\lim_\pi(x) / \operatorname(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaps Between Prime Numbers
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''''n'' every prime can be written as :p_ = 2 + \sum_^n g_i. Simple observations The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deutsche Mathematik
''Deutsche Mathematik'' (German Mathematics) was a mathematics journal founded in 1936 by Ludwig Bieberbach and Theodor Vahlen. Vahlen was publisher on behalf of the German Research Foundation (DFG), and Bieberbach was chief editor. Other editors were , Erich Schönhardt, Werner Weber (all volumes), Ernst August Weiß (volumes 1–6), , Wilhelm Süss (volumes 1–5), Günther Schulz ( de), (volumes 1–4), Georg Feigl, Gerhard Kowalewski (volumes 2–6), , Willi Rinow, (volumes 2–5), and Oswald Teichmüller (volumes 3–7). In February 1936, the journal was declared the official organ of the German Student Union (DSt) by its ''Reichsführer'', and all local DSt mathematics departments were requested to subscribe and actively contribute. In the 1940s, issues appeared increasingly delayed and bunched; the journal ended with a triple issue (due Dec 1942) in June 1944.Issue list of volume 1, 2, 3, 4, 5, 6, 7 at commons ''Deutsche Mathematik'' is also the name of a movemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Münster
Münster (; nds, Mönster) is an independent city (''Kreisfreie Stadt'') in North Rhine-Westphalia, Germany. It is in the northern part of the state and is considered to be the cultural centre of the Westphalia region. It is also a state district capital. Münster was the location of the Anabaptist rebellion during the Protestant Reformation and the site of the signing of the Treaty of Westphalia ending the Thirty Years' War in 1648. Today it is known as the bicycle capital of Germany. Münster gained the status of a ''Großstadt'' (major city) with more than 100,000 inhabitants in 1915. , there are 300,000 people living in the city, with about 61,500 students, only some of whom are recorded in the official population statistics as having their primary residence in Münster. Münster is a part of the international Euregio region with more than 1,000,000 inhabitants (Enschede, Hengelo, Gronau, Osnabrück). History Early history In 793, Charlemagne sent out Ludger as a miss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germans
, native_name_lang = de , region1 = , pop1 = 72,650,269 , region2 = , pop2 = 534,000 , region3 = , pop3 = 157,000 3,322,405 , region4 = , pop4 = 21,000 3,000,000 , region5 = , pop5 = 125,000 982,226 , region6 = , pop6 = 900,000 , region7 = , pop7 = 142,000 840,000 , region8 = , pop8 = 9,000 500,000 , region9 = , pop9 = 357,000 , region10 = , pop10 = 310,000 , region11 = , pop11 = 36,000 250,000 , region12 = , pop12 = 25,000 200,000 , region13 = , pop13 = 233,000 , region14 = , pop14 = 211,000 , region15 = , pop15 = 203,000 , region16 = , pop16 = 201,000 , region17 = , pop17 = 101,000 148,00 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bonn
The federal city of Bonn ( lat, Bonna) is a city on the banks of the Rhine in the German state of North Rhine-Westphalia, with a population of over 300,000. About south-southeast of Cologne, Bonn is in the southernmost part of the Rhine-Ruhr region, Germany's largest metropolitan area, with over 11 million inhabitants. It is a university city and the birthplace of Ludwig van Beethoven. Founded in the 1st century BC as a Roman settlement in the province Germania Inferior, Bonn is one of Germany's oldest cities. It was the capital city of the Electorate of Cologne from 1597 to 1794, and residence of the Archbishops and Prince-electors of Cologne. From 1949 to 1990, Bonn was the capital of West Germany, and Germany's present constitution, the Basic Law, was declared in the city in 1949. The era when Bonn served as the capital of West Germany is referred to by historians as the Bonn Republic. From 1990 to 1999, Bonn served as the seat of government – but no longer capital – ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cologne
Cologne ( ; german: Köln ; ksh, Kölle ) is the largest city of the German western States of Germany, state of North Rhine-Westphalia (NRW) and the List of cities in Germany by population, fourth-most populous city of Germany with 1.1 million inhabitants in the city proper and 3.6 million people in the Cologne Bonn Region, urban region. Centered on the left bank of the Rhine, left (west) bank of the Rhine, Cologne is about southeast of NRW's state capital Düsseldorf and northwest of Bonn, the former capital of West Germany. The city's medieval Catholic Cologne Cathedral (), the third-tallest church and tallest cathedral in the world, constructed to house the Shrine of the Three Kings, is a globally recognized landmark and one of the most visited sights and pilgrimage destinations in Europe. The cityscape is further shaped by the Twelve Romanesque churches of Cologne, and Cologne is famous for Eau de Cologne, that has been produced in the city since 1709, and "col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
