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Pépin's Test
In mathematics, Pépin's test is a primality test, which can be used to determine whether a Fermat number is prime. It is a variant of Proth's test. The test is named for a French mathematician, Théophile Pépin. Description of the test Let F_n=2^+1 be the ''n''th Fermat number. Pépin's test states that for ''n'' > 0, :F_n is prime if and only if 3^\equiv-1\pmod. The expression 3^ can be evaluated modulo F_n by repeated squaring. This makes the test a fast polynomial-time algorithm. However, Fermat numbers grow so rapidly that only a handful of Fermat numbers can be tested in a reasonable amount of time and space. Other bases may be used in place of 3. These bases are: :3, 5, 6, 7, 10, 12, 14, 20, 24, 27, 28, 39, 40, 41, 45, 48, 51, 54, 56, 63, 65, 75, 78, 80, 82, 85, 90, 91, 96, 102, 105, 108, 112, 119, 125, 126, 130, 147, 150, 156, 160, ... . The primes in the above sequence are called Elite primes, they are: :3, 5, 7, 41, 15361, 23041, 26881, 61441, 87041, 163841, 544001, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Criterion
In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then : a^ \equiv \begin \;\;\,1\pmod& \textx \texta\equiv x^2 \pmod,\\ -1\pmod& \text \end Euler's criterion can be concisely reformulated using the Legendre symbol: : \left(\frac\right) \equiv a^ \pmod p. The criterion first appeared in a 1748 paper by Leonhard Euler.L Euler, Novi commentarii Academiae Scientiarum Imperialis Petropolitanae, 8, 1760-1, 74; Opusc Anal. 1, 1772, 121; Comm. Arith, 1, 274, 487 Proof The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. Because the modulus is prime, Lagrange's theorem applies: a polynomial of degree can only have at most roots. In particular, has at most 2 solutions for each . This immediately implies that besides 0 there are at least distinct q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jeffrey Young .
He has written numerous books on cognitive behavioral therapy and schema therapy. His two most famous books are ''Schema Therapy'' (for professionals), and ''Reinventing Your Life'' (for the general public).
When interviewed in 2021, Jeffrey Youn ...
Jeffrey E. Young (born March 9, 1950) is an American psychologist best known for having developed schema therapy. He is the founder of the Schema Therapy Institute. After earning an undergraduate degree at Yale University, he obtained a higher education degree at the University of Pennsylvania, where he then pursued postdoctoral studies with Aaron Beck Aaron Temkin Beck (July 18, 1921 – November 1, 2021) was an American psychiatrist who was a professor in the department of psychiatry at the University of Pennsylvania. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duncan Buell
Duncan may refer to: People * Duncan (given name), various people * Duncan (surname), various people * Clan Duncan * Justice Duncan (other) Places * Duncan Creek (other) * Duncan River (other) * Duncan Lake (other), including Lake Duncan Australia *Duncan, South Australia, a locality in the Kangaroo Island Council *Hundred of Duncan, a cadastral unit on Kangaroo Island in South Australia Bahamas *Duncan Town, Ragged Island, Bahamas ** Duncan Town Airport Canada * Duncan, British Columbia, on Vancouver Island * Duncan Dam, British Columbia * Duncan City, Central Kootenay, British Columbia; see List of ghost towns in British Columbia United States * Duncan Township (other) * Duncan, Arizona * Duncan, Indiana * Duncan, Iowa * Duncan, Kentucky (other) * Duncan City, Cheboygan, Michigan * Duncan, Mississippi * Duncan, Missouri * Duncan, Nebraska * Duncan, North Carolina * Duncan, Oklahoma * Duncan, South Carolina * Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Selfridge
John Lewis Selfridge (February 17, 1927 – October 31, 2010), was an American mathematician who contributed to the fields of analytic number theory, computational number theory, and combinatorics. Education Selfridge received his Ph.D. in 1958 from the University of California, Los Angeles under the supervision of Theodore Motzkin. Career Selfridge served on the faculties of the University of Illinois at Urbana-Champaign and Northern Illinois University from 1971 to 1991 (retirement), chairing the Department of Mathematical Sciences 1972–1976 and 1986–1990. He was executive editor of Mathematical Reviews from 1978 to 1986, overseeing the computerization of its operations. He was a founder of the Number Theory Foundation, which has named its Selfridge prize in his honour. Research In 1962, he proved that 78,557 is a Sierpinski number; he showed that, when ''k'' = 78,557, all numbers of the form ''k''2''n'' + 1 have a factor in the covering set . Five ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raphael M
Raffaello Sanzio da Urbino, better known as Raphael (; or ; March 28 or April 6, 1483April 6, 1520), was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, and visual achievement of the Neoplatonic ideal of human grandeur. Together with Leonardo da Vinci and Michelangelo, he forms the traditional trinity of great masters of that period. His father was court painter to the ruler of the small but highly cultured city of Urbino. He died when Raphael was eleven, and Raphael seems to have played a role in managing the family workshop from this point. He trained in the workshop of Perugino, and was described as a fully trained "master" by 1500. He worked in or for several cities in north Italy until in 1508 he moved to Rome at the invitation of the pope, to work on the Vatican Palace. He was given a series of important commissions there and elsewhere in the city, and began to work as an architect. He was st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alfred Western
Alfred may refer to: Arts and entertainment *''Alfred J. Kwak'', Dutch-German-Japanese anime television series * ''Alfred'' (Arne opera), a 1740 masque by Thomas Arne * ''Alfred'' (Dvořák), an 1870 opera by Antonín Dvořák *"Alfred (Interlude)" and "Alfred (Outro)", songs by Eminem from the 2020 album ''Music to Be Murdered By'' Business and organisations * Alfred, a radio station in Shaftesbury, England *Alfred Music, an American music publisher *Alfred University, New York, U.S. *The Alfred Hospital, a hospital in Melbourne, Australia People * Alfred (name) includes a list of people and fictional characters called Alfred * Alfred the Great (848/49 – 899), or Alfred I, a king of the West Saxons and of the Anglo-Saxons Places Antarctica * Mount Alfred (Antarctica) Australia * Alfredtown, New South Wales * County of Alfred, South Australia Canada * Alfred and Plantagenet, Ontario * Alfred Island, Nunavut * Mount Alfred, British Columbia United States * Alfred, Maine, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James C
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus James the Just, or a variation of James, brother of the Lord ( la, Iacobus from he, יעקב, and grc-gre, Ἰάκωβος, , can also be Anglicized as " Jacob"), was "a brother of Jesus", according to the New Testament. He was an early le ... Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Quadratic Reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x^2\equiv a \bmod p for an odd prime p; that is, to determine the "perfect squares" modulo p. However, this is a non-constructive result: it gives no help at all for finding a ''specific'' solution; for this, other methods are required. For example, in the case p\equiv 3 \bmod 4 using Euler's criterion one can give an explicit formula for the "square roots" modulo p of a quadratic residue a, namely, :\pm a^ indeed, :\left (\pm a^ \right )^2=a^=a\cdot a^\equiv a\left(\frac\right)=a \bmod p. This formula only works if it is k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Symbol
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic residue (''non-residue'') is −1. Its value at zero is 0. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol. Definition Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a perfect square modulo p and is a quadratic nonresidue modulo p otherwise. The Legendre symbol is a function of a and p defined as :\left(\frac\right) = \begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Order
In number theory, given a positive integer ''n'' and an integer ''a'' coprime to ''n'', the multiplicative order of ''a'' modulo ''n'' is the smallest positive integer ''k'' such that a^k\ \equiv\ 1 \pmod n. In other words, the multiplicative order of ''a'' modulo ''n'' is the order of ''a'' in the multiplicative group of the units in the ring of the integers modulo ''n''. The order of ''a'' modulo ''n'' is sometimes written as \operatorname_n(a). Example The powers of 4 modulo 7 are as follows: : \begin 4^0 &= 1 &=0 \times 7 + 1 &\equiv 1\pmod7 \\ 4^1 &= 4 &=0 \times 7 + 4 &\equiv 4\pmod7 \\ 4^2 &= 16 &=2 \times 7 + 2 &\equiv 2\pmod7 \\ 4^3 &= 64 &=9 \times 7 + 1 &\equiv 1\pmod7 \\ 4^4 &= 256 &=36 \times 7 + 4 &\equiv 4\pmod7 \\ 4^5 &= 1024 &=146 \times 7 + 2 &\equiv 2\pmod7 \\ \vdots\end The smallest positive integer ''k'' such that 4''k'' ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3. Properties Even without knowledge that we are working in the multiplicative gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primality Test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not. Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy (its running time is polynomial in the size of the input). Some primality tests prove that a number is prime, while others like Miller–Rabin prove that a number is composite. Therefore, the latter might more accurately be called ''compositeness tests'' instead of primality tests. Simple methods The simplest primality test is ''trial division'': given an input number, ''n'', check whether it is evenly divisible by any prime number between 2 and (i.e. that the division leaves no remainder). If so, then ''n'' is composite. Otherwise, it is prime.Riesel (1994) pp.2-3 For example, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
