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There are an infinite number of primes p such that p + 2 is also prime.
Such a pair of prime numbers is called a twin prime. The conjecture has been researched by many number theorists. Mathematicians believe the conjecture to be true, based only on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes.
In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs which have a distance of 2k. The case k = 1 is the twin prime conjecture.
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result 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 for some absolute constant C > 0.
In 1940, Erdös showed that there is a constant c < 1 and infinitely many primes p such that p' − p < c ln p, where p' denotes the next prime after p. This result was successively improved; in 1986 Maier showed that a constant c < 0.25 can be used. In 2004 Goldston and Yildirim showed that c could be improved further to 0.085786...
In 1966, Chen Jingrun showed that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (i.e., the product of two primes). The approach he took involved a topic called sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjectureIn mathematics, Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states: :Every even number greater than 2 can be written as the sum of two primes. The same prime may be used twice. For example, : in similar manners.
There is also a generalization of the twin prime conjecture, known as the Hardy-Littlewood conjecture (after G. H. HardyGodfrey Harold Hardy ( February 7, 1877 December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. Non-mathematicians know him for two things: A Mathematician's Apology his essay from 19 and John LittlewoodJohn Edensor Littlewood ( June 9 1885 September 6 1977) was a British mathematician. Littlewood was born in Rochester in Kent, and studied at Cambridge University. Most of his work was in the field of mathematical analysis. He collaborated for many years), which is concerned with the distribution of twin primes, in analogy to the prime number theoremIn number theory, the prime number theorem PNT describes the approximate, asymptotic distribution of the prime numbers. For any positive real number x define : The prime number theorem then states that : where ln x is the natural logarithm of x''. This no. Let π2(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C2 as
(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that
in the sense that the quotient of the two expressions tends to 1 as x approaches infinity.
This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem. The numerical evidence behind the Hardy-Littlewood conjecture is quite impressive.
On May 26, 2004, Richard ArenstorfRichard Arenstorf is a mathematician who worked at NASA, where he received the NASA Medal for Exceptional Scientific Achievement in 1966, and retired as a full professor from Vanderbilt University. At Vanderbilt he specialized in celestial mechanics and a of Vanderbilt University submitted a 38-page proof that there are, in fact, infinitely many twin primes. On June 3, Michel Balazard of University Bordeaux reported that LemmaIn mathematics, a lemma expresses a minor theorem, of interest primarily because it serves in the proof of a major result. In a few cases, a lemma becomes well-known in its own right; this can occur when a lemma is a particularly useful restatement of ano 8 on page 35 is false.[1] As is typical in mathematical proofIn mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. Proofs employ logic but usually include some amount of natural language which of course admits some ambiguity. In fact, the vast majorits, the defect may be correctable or a substitute method may repair or replace the defect. Arenstorf withdrew his proof on June 8, noting "A serious error has been found in the paper, specifically, Lemma 8 is incorrect".
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