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Wieferich Pair
In mathematics, a Wieferich pair is a pair of prime numbers ''p'' and ''q'' that satisfy :''p''''q'' − 1 ≡ 1 (Modular arithmetic#Congruence, mod ''q''2) and ''q''''p'' − 1 ≡ 1 (mod ''p''2) Wieferich pairs are named after Germany, German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof of Mihăilescu's theorem (formerly known as Catalan's conjecture). Known Wieferich pairs There are only 7 Wieferich pairs known: :(2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence and in On-Line Encyclopedia of Integer Sequences, OEIS) Wieferich triple A Wieferich triple is a triple of prime numbers ''p'', ''q'' and ''r'' that satisfy :''p''''q'' − 1 ≡ 1 (mod ''q''2), ''q''''r'' − 1 ≡ 1 (mod ''r''2), and ''r''''p'' − 1 ≡ 1 (mod ''p''2). There are 17 known Wieferich triples: :(2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 31 ... [...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] |
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] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south; it covers an area of , with a population of almost 84 million within its 16 constituent states. Germany borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The nation's capital and most populous city is Berlin and its financial centre is Frankfurt; the largest urban area is the Ruhr. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity. A region named Germania was documented before AD 100. In 962, the Kingdom of Germany formed the bulk of the Holy Roman Empire. During the 16th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Wieferich
Arthur Josef Alwin Wieferich (April 27, 1884 – September 15, 1954) was a German mathematician and teacher, remembered for his work on number theory, as exemplified by a type of prime numbers named after him. He was born in Münster, attended the University of Münster (1903–1909) and then worked as a school teacher and tutor until his retirement in 1949. He married in 1916 and had no children. Wieferich abandoned his studies after his graduation and did not publish any paper after 1909. His mathematical reputation is founded on five papers he published while a student at Münster: *. *. *. *. *. The first three papers are related to Waring's problem. His fourth paper led to the term ''Wieferich prime'', which are p such that p^2 divides 2^(p-1) - 1." See also * Wieferich pair * Wieferich's theorem *Wieferich prime In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preda Mihăilescu
Preda V. Mihăilescu (born 23 May 1955) is a Romanian mathematician, best known for his proof of the 158-year-old Catalan's conjecture. Biography Born in Bucharest,Stewart 2013 he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving a PhD from ETH Zürich in 1997. His PhD thesis, titled ''Cyclotomy of rings and primality testing'', was written under the direction of Erwin Engeler and Hendrik Lenstra. For several years, he did research at the University of Paderborn, Germany. Since 2005, he has held a professorship at the University of Göttingen. Major research In 2002, Mihăilescu proved Catalan's conjecture.Bilu et al. 2014. This number-theoretical conjecture, formulated by the French and Belgian mathematician Eugène Charles Catalan in 1844, had stood unresolved for 158 years. Mihăilescu's proof appeared in ''Crelle's Journal ''Crelle's Journal'', or just ''Crel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mihăilescu's Theorem
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the ''only'' case of two consecutive perfect powers. That is to say, that History The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (''x'', ''y'') was restricted to be (2, 3) or (3, 2). The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case ''b'' = 2. In 1976, Robert Tijdeman applied Baker's method in transcendence theory to establish a bound on a,b and used existing results bounding ''x'',''y'' in terms of ''a'', ''b'' to give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009. Sloane is chairman of the OEIS Foundation. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 350,000 sequences, making it the largest database of its kind. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. History 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wieferich Prime
In number theory, a Wieferich prime is a prime number ''p'' such that ''p''2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime ''p'' divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians. Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the ''abc'' conjecture. , the only known Wieferich primes are 1093 and 3511 . Equivalent definitions The stronger v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Quotient
In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as= 3/ref> The smallest solutions of ''q''''p''(''a'') ≡ 0 (mod ''p'') with ''a'' = ''n'' are: :2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... {{OEIS, id=A039951 A pair (''p'', ''r'') of prime numbers such that ''q''''p''(''r'') ≡ 0 (mod ''p'') and ''q''''r''(''p'') ≡ 0 (mod ''r'') is called a Wieferich pair. References External links * Gottfried HelmsFermat-/Euler-quotients (''a''''p''-1 – 1)/''p''''k'' with arbitrary ''k'' * Richard FischerFermat quotients B^(P-1) 1 (mod P^2) Number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astérisque
'' Astérisque'' is a mathematical journal published by Société Mathématique de France and founded in 1973. It publishes mathematical monographs, conference reports, and the annual report of the Séminaire Nicolas Bourbaki The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. It is one of the major institutions of contemporary mathematics, and a baro .... External links *Astérisque – AMS Bookstore – American Mathematical Society Société Mathématique de France academic journals Mathematics journals Publications established in 1973 English-language journals Irregular journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
