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Ernst Jacobsthal
Ernst Erich Jacobsthal (16 October 1882, Berlin – 6 February 1965, Überlingen) was a German mathematician, and brother to the archaeologist Paul Jacobsthal. In 1906, he earned his PhD at the University of Berlin, where he was a student of Georg Frobenius, Hermann Schwarz and Issai Schur; his dissertation, ''Anwendung einer Formel aus der Theorie der quadratischen Reste'' (''Application of a Formula from the Theory of Quadratic Residues''), provided a proof that prime numbers of the form 4''n'' + 1 are the sum of two square numbers. In 1934, he was fired from his professorship at the Technische Hochschule Berlin, because of his Jewish origins. In 1939 he fled to Norway and became after the war a professor at the Norwegian Institute of Technology in Trondheim. See also * Jacobsthal sum * Jacobsthal number * Fermat's theorem on sums of two squares In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be express ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constituent states, Berlin is surrounded by the State of Brandenburg and contiguous with Potsdam, Brandenburg's capital. Berlin's urban area, which has a population of around 4.5 million, is the second most populous urban area in Germany after the Ruhr. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the Spree, which flows into the Havel (a tributary of the Elbe) in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree, Havel and Dahme, the largest of which is Lake Müggelsee. Due to its l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1882 Births
Year 188 (CLXXXVIII) was a leap year starting on Monday of the Julian calendar. At the time, it was known in the Roman Empire as the Year of the Consulship of Fuscianus and Silanus (or, less frequently, year 941 ''Ab urbe condita''). The denomination 188 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Publius Helvius Pertinax becomes pro-consul of Africa from 188 to 189. Japan * Queen Himiko (or Shingi Waō) begins her reign in Japan (until 248). Births * April 4 – Caracalla (or Antoninus), Roman emperor (d. 217) * Lu Ji (or Gongji), Chinese official and politician (d. 219) * Sun Shao, Chinese general of the Eastern Wu state (d. 241) Deaths * March 17 – Julian, pope and patriarch of Alexandria * Fa Zhen (or Gaoqing), Chinese scholar (b. AD 100) * Lucius Antistius Burrus, Roman politician (executed) * Ma Xiang, Chi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: :5 = 1^2 + 2^2, \quad 13 = 2^2 + 3^2, \quad 17 = 1^2 + 4^2, \quad 29 = 2^2 + 5^2, \quad 37 = 1^2 + 6^2, \quad 41 = 4^2 + 5^2. On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobsthal Number
In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal. Like the related Fibonacci numbers, they are a specific type of Lucas sequence U_n(P,Q) for which ''P'' = 1, and ''Q'' = −2—and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that. The first Jacobsthal numbers are: : 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, … A Jacobsthal prime is a Jacobsthal number that is also prime. The first Jacobsthal primes are: :3, 5, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, … Jacobsthal numbers Jacobsthal numbers are defined by the recurrence r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobsthal Sum
In mathematics, Jacobsthal sums are finite sums of Legendre symbols related to Gauss sum In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a ...s. They were introduced by . Definition The Jacobsthal sum is given by :\phi_n(a)=\sum_\left(\dfrac\right) where ''p'' is prime and () is the Legendre symbol. References * * Further reading * {{citation , last=Storer , first=Thomas , title=Cyclotomy and difference sets , location=Chicago , publisher=Markham Publishing Company , year=1967 , zbl=0157.03301 Number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trondheim
Trondheim ( , , ; sma, Tråante), historically Kaupangen, Nidaros and Trondhjem (), is a city and municipality in Trøndelag county, Norway. As of 2020, it had a population of 205,332, was the third most populous municipality in Norway, and was the fourth largest urban area. Trondheim lies on the south shore of Trondheim Fjord at the mouth of the River Nidelva. Among the major technology-oriented institutions headquartered in Trondheim are the Norwegian University of Science and Technology (NTNU), the Foundation for Scientific and Industrial Research (SINTEF), and St. Olavs University Hospital. The settlement was founded in 997 as a trading post, and it served as the capital of Norway during the Viking Age until 1217. From 1152 to 1537, the city was the seat of the Catholic Archdiocese of Nidaros; it then became, and has remained, the seat of the Lutheran Diocese of Nidaros, and the site of the Nidaros Cathedral. It was incorporated in 1838. The current municipalit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norwegian Institute Of Technology
The Norwegian Institute of Technology (Norwegian: ''Norges tekniske høgskole'', NTH) was a science institute in Trondheim, Norway. It was established in 1910, and existed as an independent technical university for 58 years, after which it was merged into the University of Trondheim as an independent college. In 1996 NTH ceased to exist as an organizational superstructure when the university was restructured and rebranded. The former NTH departments are now basic building blocks of the Norwegian University of Science and Technology (NTNU). NTH was primarily a polytechnic institute, educating master level engineers as well as architects. In 1992 NTH had 7627 master and doctoral students and 1591 employees; it graduated 1262 chartered engineers (master level), 52 chartered architects, and 92 Dr.Ing. (PhD). The operating budget was equivalent to US$100 M, and the total premises amounted to around 260,000 m2 (64 acres). Since the merger, it forms a part of the university campu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Technical University Of Berlin
The Technical University of Berlin (official name both in English and german: link=no, Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public research university located in Berlin, Germany. It was the first German university to adopt the name "Technische Universität" (Technical University). The university alumni and professor list includes several US National Academies members, two National Medal of Science laureates and ten Nobel Prize laureates. TU Berlin is a member of TU9, an incorporated society of the largest and most notable German institutes of technology and of the Top International Managers in Engineering network, which allows for student exchanges between leading engineering schools. It belongs to the Conference of European Schools for Advanced Engineering Education and Research. The TU Berlin is home of two innovation centers designated by the European Institute of Innovation and Technology. The university is labeled ... [...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] |
Überlingen
Überlingen is a German city on the northern shore of Lake Constance (Bodensee) in Baden-Württemberg near the border with Switzerland. After the city of Friedrichshafen, it is the second largest city in the Bodenseekreis (district), and a central point for the outlying communities. Since 1 January 1993, Überlingen has been categorized as a large district city (Große Kreisstadt). History The history of Überlingen dates back to Roman times, but a variety of settlements pre-dated Roman occupation. Stone age settlements, discovered along the shoreline of Lake Constance, document that the lake supported several dozen thriving communities of 50–100 individuals. These settlements fall under the category of the Hallstatt culture, and their habits, dress, and diet has been illuminated through the excavation of archaeological sites, such as a major site in Hallstadt, Austria, excavated in the mid-to late 19th century.Alfons Semler, ''Überlingen: Bilder aus der Geschichte einer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Genealogy Project
The Mathematics Genealogy Project (MGP) is a web-based database for the academic genealogy of mathematicians.. By 31 December 2021, it contained information on 274,575 mathematical scientists who contributed to research-level mathematics. For a typical mathematician, the project entry includes graduation year, thesis title (in its Mathematics Subject Classification), '' alma mater'', doctoral advisor, and doctoral students.. Origin of the database The project grew out of founder Harry Coonce's desire to know the name of his advisor's advisor.. Coonce was Professor of Mathematics at Minnesota State University, Mankato, at the time of the project's founding, and the project went online there in fall 1997.Mulcahy, Colm;The Mathematics Genealogy Project Comes of Age at Twenty-one(PDF) AMS Notices (May 2017) Coonce retired from Mankato in 1999, and in fall 2002 the university decided that it would no longer support the project. The project relocated at that time to North Dakota State U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
