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Stephens' Constant
Stephens' constant expresses the density of certain subsets of the prime numbers. Let a and b be two multiplicatively independent integers, that is, a^m b^n \neq 1 except when both m and n equal zero. Consider the set T(a,b) of prime numbers p such that p evenly divides a^k - b for some power k. The density of the set T(a,b) relative to the set of all primes is a rational multiple of : C_S = \prod_p \left(1 - \frac \right) = 0.57595996889294543964316337549249669\ldots Stephens' constant is closely related to the Artin constant C_A that arises in the study of primitive roots. :C_S= \prod_ \left( C_A + \left( \right) \right) \left( \right) See also *Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhar ... * Twin prime constant References {{numtheory-stub Algebraic nu ... [...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 alw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Number Theory
The ''Journal of Number Theory'' (''JNT'') is a bimonthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State University). It is currently published monthly by Elsevier and the editor-in-chief is Dorian Goldfeld (Columbia University). According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 0.72. References External links * Number theory Mathematics journals Publications established in 1969 Elsevier academic journals Monthly journals English-language journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Independence
In number theory, two positive integers ''a'' and ''b'' are said to be multiplicatively independent if their only common integer power is 1. That is, for integers ''n'' and ''m'', a^n=b^m implies n=m=0. Two integers which are not multiplicatively independent are said to be multiplicatively dependent. As examples, 36 and 216 are multiplicatively dependent since 36^3=(6^2)^3=(6^3)^2=216^2, whereas 6 and 12 are multiplicatively independent. Properties Being multiplicatively independent admits some other characterizations. ''a'' and ''b'' are multiplicatively independent if and only if \log(a)/\log(b) is irrational. This property holds independently of the base of the logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of .... Let a = p_1^p_2^ \cdots p_k^ and b = q_1^q_2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Constant
In number theory, Artin's conjecture on primitive roots states that a given integer ''a'' that is neither a square number nor −1 is a primitive root modulo infinitely many primes ''p''. The conjecture also ascribes an asymptotic density to these primes. This conjectural density equals Artin's constant or a rational multiple thereof. The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter's diary. The conjecture is still unresolved as of 2022. In fact, there is no single value of ''a'' for which Artin's conjecture is proved. Formulation Let ''a'' be an integer that is not a square number and not −1. Write ''a'' = ''a''0''b''2 with ''a''0 square-free. Denote by ''S''(''a'') the set of prime numbers ''p'' such that ''a'' is a primitive root modulo ''p''. Then the conjecture states # ''S''(''a'') has a positive asymptotic density inside the set of primes. In particular, ''S''(''a'') is infinite. # Under the conditions th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Root Modulo N
In modular arithmetic, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . That is, is a ''primitive root modulo'' if for every integer coprime to , there is some integer for which ≡ (mod ). Such a value is called the index or discrete logarithm of to the base modulo . So is a ''primitive root modulo'' if and only if is a generator of the multiplicative group of integers modulo . Gauss defined primitive roots in Article 57 of the '' Disquisitiones Arithmeticae'' (1801), where he credited Euler with coining the term. In Article 56 he stated that Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime . In fact, the ''Disquisitiones'' contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive. Elementary example The number 3 is a primitive root modulo 7 beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuscripta Mathematica
The Knights of Columbus Vatican Film Library in St. Louis, Missouri is the only collection, outside the Vatican itself, of microfilms of more than 37,000 works from the ''Biblioteca Apostolica Vaticana'', the Vatican Library in Europe. It is located in the Pius XII Memorial Library on the campus of Saint Louis University. History The Library was created by Lowrie J. Daly (1914–2000), with funding from the Knights of Columbus. The goal was to make Vatican and other documents more available to researchers in North America. Microfilming of Vatican manuscripts began in 1951, and according to the Library's website, was the largest microfilming project that had been undertaken up to that date. From 1951 to 1957, twelve million manuscript pages were recorded, from 30,000 different works. This represents approximately 75% of the manuscripts available in the targeted language groups. Other microfilm projects in the 1950s included Jesuit archival material from Rome, archives in both Nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. Definition In general, if is a bounded multiplicative function, then the Dirichlet series :\sum_ \frac\, is equal to :\prod_ P(p, s) \quad \text \operatorname(s) >1 . where the product is taken over prime numbers , and is the sum :\sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots In fact, if we consider these as formal generating functions, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes . An important special cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Prime Constant
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
