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Euler's Theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, then a^ is congruent to 1 modulo , where \varphi denotes Euler's totient function; that is :a^ \equiv 1 \pmod. In 1736, Leonhard Euler published a proof of Fermat's little theorem (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where is not prime. The converse of Euler's theorem is also true: if the above congruence is true, then a and n must be coprime. The theorem is further generalized by some of Carmichael's theorems. The theorem may be used to easily reduce large powers modulo n. For example, consider finding the ones place decimal digit of 7^, i.e. 7^ \pmod. The integers 7 and 10 are coprime, and \varphi( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from 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 can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lagrange's Theorem (group Theory)
In the mathematics, mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group , then , H, is a divisor of , G, , i.e. the order of a group, order (number of elements) of every subgroup H divides the order of group G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup H of a finite group G, not only is , G, /, H, an integer, but its value is the index of a subgroup, index [G:H], defined as the number of left cosets of H in G. This variant holds even if G is infinite, provided that , G, , , H, , and [G:H] are interpreted as cardinal numbers. Proof The left cosets of in are the equivalence classes of a certain equivalence relation on : specifically, call and in equivalent if there exists in such that . Therefore, the set of left cosets forms a Partition of a set, partition of . Each left coset has the same cardinality as because x \mapsto ax defines a bijection H \to aH ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Theorems In Number Theory
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, 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 example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Springer Publishing
Springer Publishing Company is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology). It was established in 1951 by Bernhard Springer, a great-grandson of Julius Springer, and is based in Midtown Manhattan, New York City. History Springer Publishing Company was founded in 1950 by Bernhard Springer, the Berlin-born great-grandson of Julius Springer, who founded Springer Science+Business Media, Springer-Verlag (now Springer Science+Business Media). Springer Publishing's first landmark publications included ''Livestock Health Encyclopedia'' by R. Seiden and the 1952 ''Handbook of Cardiology for Nurses''. The company's books soon branched into other fields, including medicine and psychology. Nursing publications grew rapidly in number, as Modell's ''Drugs in Current Use'', a small annual paperback, sold over 150,000 copies over several edi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Disquisitiones Arithmeticae
(Latin for ''Arithmetical Investigations'') is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory. In this book, Gauss brought together and reconciled results in number theory obtained by such eminent mathematicians as Fermat, Euler, Lagrange, and Legendre, while adding profound and original results of his own. Scope The ''Disquisitiones'' covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. Gauss did not explicitly recognize the concept of a group, which is central to modern algebra, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the ''Disquisitiones'', Gauss describes the scope of the book as follows: Gauss also writes, "When confronting man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Wilson's Theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of modular arithmetic), the factorial (n - 1)! = 1 \times 2 \times 3 \times \cdots \times (n - 1) satisfies :(n-1)!\ \equiv\; -1 \pmod n exactly when ''n'' is a prime number. In other words, any integer ''n'' > 1 is a prime number if, and only if, (''n'' − 1)! + 1 is divisible by ''n''. History The theorem was first stated by Ibn al-Haytham . Edward Waring announced the theorem in 1770 without proving it, crediting his student John Wilson for the discovery. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but never published it. Example For each of the values of ''n'' from 2 to 30, the following table shows the number (''n'' −&thinsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 \textx^2\equiv a \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 dates from 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 quadrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Carmichael Number
In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation: : b^n\equiv b\pmod for all integers . The relation may also be expressed in the form: : b^\equiv 1\pmod for all integers b that are relatively prime to . They are infinite set, infinite in number. They constitute the comparatively rare instances where the strict converse of Fermat's Little Theorem does not hold. This fact precludes the use of that theorem as an absolute test of Prime numbers, primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. The Carmichael numbers were named after the American mathematician Robert Daniel Carmichael, Robert Carmichael by N. G. W. H. Beeger, Nicolaas Beeger, in 1950. Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short. Overview Fermat's little theorem states that if p is a prime number, then for any integer , the number b^p-b is an i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |