A twin prime is a prime number that differs from another prime number by two. Except for the pair (2, 3), this is the smallest possible difference between two primes. Some examples of twin prime pairs are 5 and 7, 11 and 13, and 821 and 823. (Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin.) In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... 2 (two) is the natural number following 1 and preceding 3. ...

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. A strong form of the twin prime conjecture, the Hardy-Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem. Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

Using his celebrated sieve method, Viggo Brun shows that the number of twin primes less than x is << x/(log x)². This result implies that the sum of the reciprocals of all twin primes converges (see Brun's constant and Brun's theorem). This is in stark contrast to the sum of the reciprocals of all primes, which diverges. He also shows that every even number can be represented in infinitely many ways as a difference of two numbers both having at most 9 prime factors. Chen Jingrun's well known theorem states that for any m even, there are infinitely many primes that differ by m from a number having at most two prime factors. (Before Brun attacked the twin prime problem, Jean Merlin (1876-1914) had also attempted to solve this problem using the sieve method. He was killed in World War I.) Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. ... Viggo Brun (October 13, 1882 - August 15, 1978) was a Norwegian mathematician. ... In mathematics, a series is a sum of a sequence of terms. ... The reciprocal function: y = 1/x. ... In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a mathematical constant now called Bruns constant for twin primes and usually denoted by B2 (sequence A065421 in OEIS): in stark contrast to the... In mathematics, Bruns theorem is a result of Viggo Brun in number theory. ... Chen Jingrun (ch. ... Combatants Allied Powers: Russian Empire France British Empire Italy United States Central Powers: Austria-Hungary German Empire Ottoman Empire Bulgaria Commanders Nicholas II Aleksei Brusilov Georges Clemenceau Joseph Joffre Ferdinand Foch Robert Nivelle Herbert Henry Asquith Sir Douglas Haig Sir John Jellicoe Victor Emmanuel III Luigi Cadorna Armando Diaz Woodrow...

Every twin prime pair greater than 3 is of the form (6n − 1, 6n + 1) for some natural number n, and with the exception of n = 1, n must end in 0, 2, 3, 5, 7, or 8. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

It has been proven that the pair m, m + 2 is a twin prime if and only if

Largest

On January 15, 2007 two distributed computing projects, Twin Prime Search and PrimeGrid found the largest known twin primes, 2003663613 · 2¹⁹⁵⁰⁰⁰±1. The numbers have 58711 decimal digits. The discoverer was Eric Vautier (France). January 15 is the 15th day of the year in the Gregorian calendar. ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the Anno Domini (common) era. ... Twin Prime Search (TPS) is a distributed computing project that looks for large twin primes. ... PrimeGrid is a distributed computing project for factoring the number in the RSA Factoring Challenge, and also to test PerlBOINC. It makes use of the Berkeley Open Infrastructure for Network Computing (BOINC) platform. ... The decimal (base ten or occasionally denary) numeral system has ten as its base. ... In mathematics and computer science, a numerical digit is a symbol, e. ...

An empirical analysis of all prime pairs up to 4.35 · 10¹⁵ shows that the number of such pairs less than x is x·f(x)/(log x)² where f(x) is about 1.7 for small x and decreases to about 1.3 as x tends to infinity. The limiting value of f(x) is conjectured to equal the twin prime constant The twin prime conjecture is a famous problem in number theory that involves prime numbers. ...

this conjecture would imply the twin prime conjecture, but remains unresolved.

The first 35 twin prime pairs

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463), (521, 523), (569, 571), (599, 601), (617, 619), (641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883)

Every third odd number is divisible by 3, so 5 is the only prime which is part of two pairs. The lower member of a pair is by definition a Chen prime. A prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (that is, if ,where is the Big Omega function. ...

See also

• Prime number
• Cousin prime
• Sexy prime
• Prime quadruplet
• Factorial
• Modular arithmetic

In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, a cousin prime is a pair of prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six. ... In mathematics, a sexy prime is a pair of prime numbers that differ by six; compare this with twin primes, pairs of prime numbers that differ by two, and cousin primes, pairs of prime numbers that differ by four. ... A prime quadruplet is a group of four primes, consisting of two pairs of twin primes separated only by three non-primes, specifically, a multiple of 2, a multiple of 15 and another multiple of 2. ... The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a number n is the product of all positive integers less than or equal to n. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

External links

• Top-20 Twin Primes at Chris Caldwell's Prime Pages.
• Twin Prime Search, a distributed computing project.
• Xavier Gourdon, Pascal Sebah: Introduction to Twin Primes and Brun's Constant
• Twin Prime Search and PrimeGrid finds world's largest twin primes

Primegrid 16 January 2007. Accessed January 2007. The Prime pages is a website about prime numbers maintained by Prof. ...

• Eric W. Weisstein, Twin Primes at MathWorld.