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Robert Daniel Carmichael
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician. Biography Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was studying towards his Ph.D. degree at Princeton University. Carmichael completed the requirements for his Ph.D. in mathematics in 1911. Carmichael's Ph.D. research in mathematics was done under the guidance of the noted American mathematician G. David Birkhoff, and it is considered to be the first significant American contribution to the knowledge of differential equations in mathematics. Carmichael next taught at Indiana University from 1911 to 1915. Then he moved on to the University of Illinois, where he remained from 1915 until his retirement in 1947. Carmichael is known for his research in what are now called the Carmichael numbers (a subset of Fermat pseudoprimes, numbers satisfying properties of primes described by Fermat's Little ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goodwater, Alabama
Goodwater is a town in Coosa County, Alabama, United States. At the 2020 census, the population was 1,291. It is part of the Talladega-Sylacauga Micropolitan Statistical Area. Geography Goodwater is located near the northeast corner of Coosa County at . According to the U.S. Census Bureau, the city has a total area of , of which , or 0.45%, is water. Demographics 2020 census As of the 2020 United States census, there were 1,291 people, 484 households, and 230 families residing in the town. 2010 census At the 2010 census there were 1,475 people, 618 households, and 394 families living in the city. The population density was . There were 708 housing units at an average density of . The racial makeup of the city was 73.7% Black or African American, 24.3% White, 0.7% Native American, 0.0% Asian, 0.7% from other races, and 0.5% from two or more races. 0.7% of the population were Hispanic or Latino of any race. The age distribution was 22.2% under the age of 18, 6.8% from 18 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carmichael's Theorem
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind ''U''''n''(''P'', ''Q'') with relatively prime parameters ''P'', ''Q'' and positive discriminant, an element ''U''''n'' with ''n'' ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = ''U''12(1, −1) = 144 and its equivalent ''U''12(−1, −1) = −144. In particular, for ''n'' greater than 12, the ''n''th Fibonacci number F(''n'') has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof. Statement Given two relatively prime integers ''P'' and ''Q'', such that D=P^2-4Q>0 and , let be the Lucas sequence of the first kind defined by :\begin U_0(P,Q)&=0, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theorists
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, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoprimes
A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy. Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes. Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically lower but still prohibitively high). But finding two large prime numbers as needed for this use is also expensive, so various probabilistic primality tests are used, some of which in rare cases inappropriately deliver composite numbers instead of primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lincoln La Paz
Lincoln LaPaz (February 12, 1897 – October 19, 1985) was an American astronomer from the University of New Mexico and a pioneer in the study of meteors. Early life and education He was born in Wichita, Kansas on February 12, 1897 to Charles Melchior LaPaz and Emma Josephine (Strode). He earned his Bachelor of Arts in 1920 in mathematics at Fairmont College (presently Wichita State University) and also taught there between 1917 and 1920. He earned his master's degree via a scholarship at Harvard University, completed in 1922. On June 18, 1922, he married Leota Ray Butler and later had two children, Leota Jean and Mary Strode. Between 1922 and 1925 he taught at Dartmouth College. He received his Ph.D. in 1928 at the University of Chicago, where he instructed for a short time and acted as National Research Fellow. In 1930, he was assistant professor at Ohio State University and became associate professor in 1936 and finally professor in 1942, where he helped develop the graduat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Henry Weaver
James Henry Weaver (10 June 1883 in Madison County, Ohio – 7 April 1942 in Franklin County, Ohio) was an American mathematician. Weaver received B.A. in 1908 from Otterbein College and M.A. in 1911 from Ohio State University. He was a teaching assistant at Ohio State University from 1910 to 1912. He entered the mathematics doctoral program at the University of Pennsylvania in 1912 and graduated there in 1916 with advisor Maurice Babb and thesis ''Some Extensions of the Work of Pappus and Steiner on Tangent Circles''. From 1912 to 1917 he was head of the mathematics department of West Chester High School in West Chester, Pennsylvania. He became an instructor in 1917 and in 1920 an assistant professor at Ohio State University. He was an Invited Speaker of the ICM in 1924. Selected publications * (See Pappus of Alexandria.) * (See angle trisection.) * (See Platonic solid.) * (See doubling the cube.) * * * * * * (See Steiner chain.) * (See strophoid.) * *with R. D. Carmic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Theory Of Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of special relativity, postulates: # The laws of physics are Invariant (physics), invariant (that is, identical) in all Inertial frame of reference, inertial frames of reference (that is, Frame of reference, frames of reference with no acceleration). # The speed of light in vacuum is the same for all observers, regardless of the motion of the light source or the observer. Origins and significance Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "Annus Mirabilis papers#Special relativity, On the Electrodynamics of Moving Bodies".Albert Einstein (1905)''Zur Elektrodynamik bewegter Körper'', ''Annalen der Physik'' 17: 891; English translatioOn the Electrodynamics of Moving Bodies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the family to China to work as missionaries, and he did not return to Europe until he was nine. After his schooling, Witt went to the University of Freiburg and the University of Göttingen. He joined the NSDAP (Nazi Party) and was an active party member. Witt was awarded a Ph.D. at the University of Göttingen in 1934 with a thesis titled: "Riemann-Roch theorem and zeta-Function in hypercomplexes" (Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen) that was supervised by Gustav Herglotz with Emmy Noether suggesting the top for the doctorate. He qualified to become a lecturer and gave guest lectures in Göttingen and Hamburg. He became associated with the team led by Helmut Hasse who led his habilitation. In June 1936 gave his habil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner System
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
