Twin prime

A twin prime is a prime number, either 2 less or 2 more than another prime number—for example, either member of the twin prime pair. In other words, a twin prime is a prime. Sometimes the term twin prime is used for a pair of twin primes. Twin primes become 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 or there is a largest pair; the 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. The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years; this is the content of the twin prime conjecture, which states that there are infinitely many primes p such that p + 2 is prime. In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is prime.

The case k = 1 of de Polignac's conjecture is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem. On April 17, 2013, Yitang Zhang announced a proof that for some integer N, less than 70 million, there are infinitely many pairs of primes that differ by N. Zhang's paper was accepted by Annals of Mathematics in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound; as of April 14, 2014, one year after Zhang's announcement, the bound has been reduced to 246. Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that the bound has been reduced to 12 and 6, respectively; these improved bounds were discovered using a different approach, simpler than Zhang's and was discovered independently by James Maynard and Terence Tao. This second approach gave bounds for the smallest f needed to guarantee that infinitely many intervals of width f contain at least m primes.

The pair is not considered to be a pair of twin primes. Since 2 is the only prime, this pair is the only pair of prime numbers that differ by one; the first few twin prime pairs are:, … OEIS: A077800. Five is the only prime; every twin prime pair except is of the form for some natural number n. As a result, the sum of any pair of twin primes is divisible by 12. In 1915, Viggo Brun showed; this famous result, called Brun's theorem, was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed C N 2 for some absolute constant C > 0. In fact, it is bounded above by: C ′ N 2, where C ′ = 8 C 2, where C2 is the twin prime constant, given below. In 1940, Paul Erdős showed that there is a constant c < 1 and infinitely many primes p such that < where p′ denotes the next prime after p. What this means is that we can find infinitely many intervals that contain two primes as long as we let these intervals grow in size as we move to bigger and bigger primes.

Here, "grow slowly" means that the length of these intervals can grow logarithmically. This result was successively improved. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786… In 2005, Goldston, János Pintz and Yıldırım established that c can be chosen to be arbitrarily small, i.e. lim inf n → ∞ p n + 1 − p n log ⁡ p n = 0. On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example, c ln ln p. By assuming the Elliott–H


The City municipality of Sevojno is a town and one of two city municipalities which constitute the City of Užice. As of 2011, it has 7,101 inhabitants. In February 2012, on a non-binding referendum, the citizens of Sevojno voted for the creation of a separate city municipality within the City of Užice. However, residents of the nearby villages of Zlakusa and Krvavci did not support the initiative. Assembly of the City supported formation of the municipality. In 2013, the city municipality of Sevojno was established; as of October 2013 the budget negotiations are under way. Sevojno is an industrial center of Užice, with aluminum mill Impol Seval, copper mill Valjaonica bakra Sevojno,construction company MPP Jedinstvo and metal recycling facility "Inos-Sinma". Sevojno was the home to the former football club FK Sevojno. In 2010, it merged with FK Sloboda Užice. Official website

James Ford Bell Library

The James Ford Bell Library is a special collection of the University of Minnesota Libraries located on the University of Minnesota Minneapolis campus. It is named for its donor and patron James Ford Bell, founder of the General Mills Corporation in Minneapolis, Minnesota; the collection consists of some 30,000 rare books, manuscripts, broadsides and other materials documenting the history and impact of international trade in the pre-modern era, before ca. 1800. Its materials range in date from 400 CE to 1825 CE, with the bulk of the collection concentrated between the years 1450 and 1790, the early modern period; the library is known for its globe gores copy of the 1507 Waldseemuller world map, it acquired a copy of the 1602 Impossible Black Tulip Chinese world map in 2009. The scope of the collection is global and more than 15 languages are represented; the library was founded at the University of Minnesota in 1953 and is located in the university's Wilson Library building. The Associates of the James Ford Bell Library was established in 1963 as friends group that contributes to the support of the library and sponsors events and publications.

The library has a variety of publications and since 1964 has sponsored an annual public lecture series: the James Ford Bell Lecture. Dr. John "Jack" Parker, 1953-1991 Dr. Carol Urness, 1991-2001 Dr. Brian Fryckenberg, 2003 Dr. Marguerite Ragnow, 2005–present The James Ford Bell Library: An annotated catalog of original source materials relating to the history of European expansion, 1400-1800 Minneapolis, Minn.: James Ford Bell Library, University of Minnesota, 1994. James Ford Bell and his books: the nucleus of a library. Minneapolis, Minn.: Associates of the James Ford Bell Library, University of Minnesota, 1993. A book for Jack: words to, by and about John Parker, curator of the James Ford Bell Library, University of Minnesota, edited by Carol Urness. Minneapolis/St. Paul: Associates of the James Ford Bell Library, 1991; the world for a marketplace: episodes in the history of European expansion: commemorating the 25th anniversary of the James Ford Bell Library, by John Parker. Minneapolis: Associates of the James Ford Bell Library, 1978.

The Manifest: a newsletter to the Associates of the James Ford Bell Library, Wilson Library, University of Minnesota. The merchant explorer: a commentary on selected recent acquisitions. 1961- James Ford Bell Library