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Andrica's Conjecture
Andrica's conjecture (named afteDorin Andrica is a conjecture regarding the gaps between prime numbers. The conjecture states that the inequality :\sqrt - \sqrt < 1 holds for all , where is the ''n''th prime number. If denotes the ''n''th , then Andrica's conjecture can also be rewritten as : Empirical evidence Imran Ghory has used data on the largest prime gaps to confirm the conjecture for up to 1.3002 × 1016.''Prime Numbers: The Most Mysterious Figures in Math'', John Wiley & Sons, Inc., 2005, p. 13. Using a table of maximal gaps and the above gap inequality, the con ...[...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] |
Prime Gap
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''''n'' every prime can be written as :p_ = 2 + \sum_^n g_i. Simple observations The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ' ... [...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 always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Gaps
A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g''''n'' or ''g''(''p''''n'') is the difference between the (''n'' + 1)-th and the ''n''-th prime numbers, i.e. :g_n = p_ - p_n.\ We have ''g''1 = 1, ''g''2 = ''g''3 = 2, and ''g''4 = 4. The sequence (''g''''n'') of prime gaps has been extensively studied; however, many questions and conjectures remain unanswered. The first 60 prime gaps are: :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, ... . By the definition of ''g''''n'' every prime can be written as :p_ = 2 + \sum_^n g_i. Simple observations The first, smallest, and only odd prime gap is the gap of size 1 between 2, the only even prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotically
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Andrica's Conjecture
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramér's Conjecture
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that :p_-p_n=O((\log p_n)^2),\ where ''p''''n'' denotes the ''n''th prime number, ''O'' is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his heuristic actually supports the stronger statement :\limsup_ \frac = 1, and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven. Conditional proven results on prime gaps Cramér gave a conditional proof of the much weaker statement that :p_-p_n = O(\s ... [...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] |
Firoozbakht's Conjecture
In number theory, Firoozbakht's conjecture (or the Firoozbakht conjecture) is a conjecture about the distribution of prime numbers. It is named after the Iranian mathematician Farideh Firoozbakht who stated it first in 1982. The conjecture states that p_^ (where p_n is the ''n''th prime) is a strictly decreasing function of ''n'', i.e., :\sqrt +10, where \gamma denotes the Euler–Mascheroni constant. Two related conjectures (see the comments of ) are :\left(\frac\right)^n < e, which is weaker, and : which is stronger. See also * * Andrica's conjecture * |
Springer-Verlag
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, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PlanetMath
PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, "PlanetMath.org, Ltd". PlanetMath was started when the popular free online mathematics encyclopedia MathWorld was temporarily taken offline for 12 months by a court injunction as a result of the CRC Press lawsuit against the Wolfram Research company and its employee (and MathWorld's author) Eric Weisstein. Materials The main PlanetMath focus is on encyclopedic entries. It formerly operated a self-hosted forum, but now encourages discussion via Gitter. , the encyclopedia hosted about 9,289 entries and over 16,258 ''concepts'' (a concept may be for example a specific notion defined within a more general entry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
