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Mills' Constant
In number theory, Mills' constant is defined as the smallest positive real number ''A'' such that the floor function of the double exponential function : \lfloor A^ \rfloor is a prime number for all natural numbers ''n''. This constant is named after William Harold Mills who proved in 1947 the existence of ''A'' based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... . Mills primes The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins :2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, \ldots . If ''ai'' denotes the ''i'' th prime in this sequence, then ''ai'' can be calculated as the smallest prime number larger than a_^3. In order to ensure that rounding A^, for ''n'' = 1, 2, 3, & ... [...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] |
Probable Prime
In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different specific conditions. While there may be probable primes that are composite (called pseudoprimes), the condition is generally chosen in order to make such exceptions rare. Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer ''n'', choose some integer ''a'' that is not a multiple of ''n''; (typically, we choose ''a'' in the range ). Calculate . If the result is not 1, then ''n'' is composite. If the result is 1, then ''n'' is likely to be prime; ''n'' is then called a probable prime to base ''a''. A weak probable prime to base ''a'' is an integer that is a probable prime to base ''a'', but which is not a strong probable prime to base ''a'' (see below). For a fixed base ''a'', it is unusual f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *''Memoirs of the American Mathematical Society'' *''Notices of the American Mathematical Society'' *'' Proceedings of the American M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formula For Primes
In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. Formulas based on Wilson's theorem A simple formula is :f(n) = \left\lfloor \frac \right\rfloor (n-1) + 2 for positive integer n, where \lfloor\ \rfloor is the floor function, which rounds down to the nearest integer. By Wilson's theorem, n+1 is prime if and only if n! \equiv n \pmod. Thus, when n+1 is prime, the first factor in the product becomes one, and the formula produces the prime number n+1. But when n+1 is not prime, the first factor becomes zero and the formula produces the prime number 2. This formula is not an efficient way to generate prime numbers because evaluating n! \bmod (n+1) requires about n-1 multiplications and reductions modulo n+1. In 1964, Willans gave the formula :p_n = 1 + \sum_^ \left\lfloor \left(\fra ... [...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] |
Ceiling Function
In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, , , , and . Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). Some authors may define the integral part as if is nonnegative, and otherwise: for example, and . The operation of truncation generalizes this to a specified number of digits: truncation to zero significant digits is the same as the integer part. For an integer, . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl Friedrich Gauss introduced the square bracket notation in his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; , the conjecture has neither been proved nor disproved. Prime gaps If Legendre's conjecture is true, the gap between any prime ''p'' and the next largest prime would be O(\sqrt p), as expressed in big O notation. It is one of a family of results and conjectures related to prime gaps, that is, to the spacing between prime numbers. Others include Bertrand's postulate, on the existence of a prime between n and 2n, Oppermann's conjecture on the existence of primes between n^2, n(n+1), and (n+1)^2, Andrica's conjecture and Brocard's conjecture on the existence of primes between squares of consecutive primes, and Cramér's conjecture that the gaps are always much smaller, of the order (\log p)^2. If Cramér's conjecture is true, Legendre's conjecture would follow for all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression (mathematics), infinite expression. In either case, all integers in the sequence, other than the first, must be positive number, positive. The integers a_i are called the coefficients or terms of the continued fraction. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or function (mathematics), functions are used in place of one or more of the numerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Provable Prime
In number theory, a provable prime is an integer that has been calculated to be prime using a primality-proving algorithm. Boot-strapping techniques using Pocklington primality test are the most common ways to generate provable primes for cryptography. Contrast with probable prime, which is likely (but not certain) to be prime, based on the output of a probabilistic primality test. In principle, every prime number can be proved to be prime in polynomial time by using the AKS primality test. Other methods which guarantee that their result is prime, but which do not work for all primes, are useful for the random generation of provable primes. Provable primes have also been generated on embedded devices. See also *Primality test *Probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Different types of probable primes have different spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
