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Carmichael Number
In number theory, a Carmichael number is a composite number n, which in modular arithmetic satisfies the congruence relation: :b^n\equiv b\pmod for all integers b. The relation may also be expressed in the form: :b^\equiv 1\pmod. for all integers b which are relatively prime to n. Carmichael numbers are named after American mathematician Robert Carmichael, the term having been introduced by Nicolaas Beeger in 1950 (Øystein Ore had referred to them in 1948 as numbers with the "Fermat property", or "''F'' numbers" for short). They are 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 primality. The Carmichael numbers form the subset ''K''1 of the Knödel numbers. Overview Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''b'', the number ''b'' − ''b'' is an integer multipl ... [...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 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 object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Pseudoprime
A strong pseudoprime is a composite number that passes the Miller–Rabin primality test. All prime numbers pass this test, but a small fraction of composites also pass, making them "pseudoprimes". Unlike the Fermat pseudoprimes, for which there exist numbers that are pseudoprimes to all coprime bases (the Carmichael numbers), there are no composites that are strong pseudoprimes to all bases. Motivation and first examples Let us say we want to investigate if ''n'' = 31697 is a probable prime (PRP). We pick base ''a'' = 3 and, inspired by Fermat's little theorem, calculate: : 3^ \equiv 1 \pmod This shows 31697 is a Fermat PRP (base 3), so we may suspect it is a prime. We now repeatedly halve the exponent: : 3^ \equiv 1 \pmod : 3^ \equiv 1 \pmod : 3^ \equiv 28419 \pmod The first couple of times do not yield anything interesting (the result was still 1 modulo 31697), but at exponent 3962 we see a result that is neither 1 nor minus 1 (i.e. 31696) modulo 31697. This proves 31697 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dickson's Conjecture
In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by that for a finite set of linear forms , , ..., with , there are infinitely many positive integers for which they are all prime, unless there is a congruence condition preventing this . The case ''k'' = 1 is Dirichlet's theorem. Two other special cases are well-known conjectures: there are infinitely many twin primes (''n'' and 2 + ''n'' are primes), and there are infinitely many Sophie Germain primes (''n'' and 1 + 2''n'' are primes). Dickson's conjecture is further extended by Schinzel's hypothesis H. Generalized Dickson's conjecture Given ''n'' polynomials with positive degrees and integer coefficients (''n'' can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime ''p'' there is an integer ''x'' such that the values of all ''n'' polynomials at ''x'' are not divisible by ''p'', then there are i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Václav Šimerka
Václav Šimerka (20 December 1819 – 26 December 1887) was a Bohemian mathematician, priest, physicist, and philosopher. He wrote the first Czech text on calculus and is credited for discovering the first seven Carmichael numbers, from 561 to 8911, in 1885. Biography Šimerka was born on 20 December 1819 in Vysoké Veselí in Bohemia to a family of coopers of businessman Petr Šimerka and his wife Terezie. After attending school in Jičín, he studied in the University of Prague's Faculty of Philosophy from 1839 to 1841. There, he studied mathematics under Jakob Philipp Kulik and astronomy under and practical geometry under Adam Bittner and also obligatory teaching of religion, philosophy, mathematics, Latin philology, natural science, physics, moral philosophy and history. After graduating in Prague, Šimerka studied in the Theological Seminary in Hradec Králové. Šimerka was ordained on 25 July 1845 and then became a chaplain in Žlunice near Jičín. He only spent a sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Number (group Theory)
A cyclic number is a natural number ''n'' such that ''n'' and φ(''n'') are coprime. Here φ is Euler's totient function. An equivalent definition is that a number ''n'' is cyclic if and only if any group of order ''n'' is cyclic. Any prime number is clearly cyclic. All cyclic numbers are square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ....For if some prime square ''p''2 divides ''n'', then from the formula for φ it is clear that ''p'' is a common divisor of ''n'' and φ(''n''). Let ''n'' = ''p''1 ''p''2 … ''p''''k'' where the ''p''''i'' are distinct primes, then φ(''n'') = (''p''1 − 1)(''p''2 − 1)...(''p''''k'' – 1). If no ''p''''i'' divides any (''p''''j'' – 1), then ''n'' and φ(''n'') have no common (prime) divisor, and ''n'' is cyclic. The first cyclic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Number
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Divisor
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] |
Square-free Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alwin Korselt
Alwin Reinhold Korselt (17 March 1864, in Mittelherwigsdorf – 4 February 1947, in Plauen) was a German mathematician. He discovered Korselt's criterion, which provides a secondary definition for Carmichael numbers and also contributed an early result in algebraic logic. Personal life The Korselts are a huge, widespread family that has been resident in the village of Mittelherwigsdorf near Zittau in Saxony (nowadays close to the Czech and Polish borders) since the early Middle Ages. Alwin Korselt was born there in 1864. After attending '' Gymnasium'' in Zittau between 1876 and 1885, he studied mathematics and physics in Leipzig (with one semester in 1886 in Freiburg im Breisgau) until 1890. After a probationary year as a teacher at Nikolaigymnasium in Leipzig he taught from 1891 to 1898 at various schools in Pirna, Dresden, Keilhau near Rudolstadt, Löbau and Meerane. Frequent changes, often between very different school types, indicate that he found it difficult to gain groun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
