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Chinese Hypothesis
In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer ''n'' is prime if and only if it satisfies the condition that 2^n-2 is divisible by ''n''—in other words, that an integer ''n'' is prime if and only if 2^n \equiv 2 \bmod. It is true that if ''n'' is prime, then 2^n \equiv 2 \bmod (this is a special case of Fermat's little theorem), however the converse (if 2^n \equiv 2 \bmod then ''n'' is prime) is false, and therefore the hypothesis as a whole is false. The smallest counterexample is ''n'' = 341 = 11×31. Composite numbers ''n'' for which 2^n-2 is divisible by ''n'' are called Poulet numbers. They are a special class of Fermat pseudoprimes. History Once, and sometimes still, mistakenly thought to be of ancient Chinese origin, the Chinese hypothesis actually originates in the mid-19th century from the work of Qing dynasty mathematician Li Shanlan Li Shanlan (李善蘭, courtesy name: Renshu 壬叔, art name: Qiuren 秋紉) (1810 – 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
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 arithmetic function, 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 Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poulet Number
Poulet is a French surname, meaning chicken. Notable people with the name include: * Anne Poulet (born 1942), American art historian * Gaston Poulet (1892–1974), French violinist and conductor * Georges Poulet (1902–1991), Belgian literary critic * J. Poulet (fl. 1811–1818), English cricketer * Olivia Poulet (born 1978), English actress and screenwriter * Paul Poulet (1887–1946), Belgian mathematician * Quentin Poulet (fl. 1477–1506), Burgundian Catholic priest, scribe, illuminator, and librarian * Robert Poulet (1893–1989), Belgian writer, literary critic and journalist * William Poulet (publisher), pseudonym used by Jean-Paul Wayenborgh to write his History of Spectacles "Die Brille" * Auguste Poulet-Malassis (1825–1878), French printer and publisher See also * Poulett Poulett is a surname and given name. Notable people with the name include: Surname * Anne Poulett (1711–1785), fourth son of John Poulett, 1st Earl Poulett, was a British Member of Parliament * G ... [...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] |
American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of The Theory Of Numbers
''History of the Theory of Numbers'' is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written . Volumes * Volume 1 - Divisibility and Primality - 486 pages * Volume 2 - Diophantine Analysis In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ... - 803 pages * Volume 3 - Quadratic and Higher Forms - 313 pages References * * * * * * * * * * * * External links History of the Theory of Numbers - Volume 1at the Internet Archive. History of the Theory of Numbers - Volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Li Shanlan
Li Shanlan (李善蘭, courtesy name: Renshu 壬叔, art name: Qiuren 秋紉) (1810 – 1882) was a Chinese mathematician of the Qing Dynasty. A native of Haining, Zhejiang, he was fascinated by mathematics since childhood, beginning with the ''Nine Chapters on the Mathematical Art''. He eked out a living by being a private tutor for some years before fleeing to Shanghai in 1852 to evade the Taiping Rebellion. There he collaborated with Alexander Wylie, Joseph Edkins and others to translate many Western mathematical works into Chinese, including ''Elements of Analytical Geometry and of the Differential and Integral Calculus'' by Elias Loomis, Augustus De Morgan's ''Elements of Algebra'', and the last nine volumes of ''Euclid's Elements'' (from Henry Billingsley's edition), the first six volumes of which having been rendered into Chinese by Matteo Ricci and Xu Guangqi in 1607. A great number of mathematical terms used in Chinese today were first coined by Li, who were later ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Qing Dynasty
The Qing dynasty ( ), officially the Great Qing,, was a Manchu-led imperial dynasty of China and the last orthodox dynasty in Chinese history. It emerged from the Later Jin dynasty founded by the Jianzhou Jurchens, a Tungusic-speaking ethnic group who unified other Jurchen tribes to form a new "Manchu" ethnic identity. The dynasty was officially proclaimed in 1636 in Manchuria (modern-day Northeast China and Outer Manchuria). It seized control of Beijing in 1644, then later expanded its rule over the whole of China proper and Taiwan, and finally expanded into Inner Asia. The dynasty lasted until 1912 when it was overthrown in the Xinhai Revolution. In orthodox Chinese historiography, the Qing dynasty was preceded by the Ming dynasty and succeeded by the Republic of China. The multiethnic Qing dynasty lasted for almost three centuries and assembled the territorial base for modern China. It was the largest imperial dynasty in the history of China and in 1790 the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Pseudoprime
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. Definition Fermat's little theorem states that if ''p'' is prime and ''a'' is coprime to ''p'', then ''a''''p''−1 − 1 is divisible by ''p''. For an integer ''a'' > 1, if a composite integer ''x'' divides ''a''''x''−1 − 1, then ''x'' is called a Fermat pseudoprime to base ''a''. In other words, a composite integer is a Fermat pseudoprime to base ''a'' if it successfully passes the Fermat primality test for the base ''a''. The false statement that all numbers that pass the Fermat primality test for base 2, are prime, is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2. Pseudoprimes to base 2 are sometimes called Sarrus numbers, afte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Composite Number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are: :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a counterexample to the generalization “students are lazy”, and both a counterexample to, and disproof of, the universal quantification “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done by considering a case where a part of the hypothesis is not satisfied and the conclusion of the theorem does not hold. In mathematics In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Converse (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposition ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse". Implicational converse Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the converse of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse, unless the antecedent ''P'' and the consequent ''Q'' are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
