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42-583: Twin primes become increasingly 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 (the so-called twin prime conjecture ) or if there is a largest pair. The breakthrough 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. Usually

84-467: A Chen prime . If m  − 4 or m  + 6 is also prime then the three primes are called a prime triplet . It has been proven that the pair ( m ,  m  + 2) is a twin prime if and only if For a twin prime pair of the form (6 n − 1, 6 n + 1) for some natural number n > 1, n must end in the digit 0, 2, 3, 5, 7, or 8 ( OEIS :  A002822 ). If n were to end in 1 or 6, 6 n would end in 6, and 6 n  −1 would be

126-597: A 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor [Tzuong-Tsieng Moh] did not write me letters of recommendation." Zhang made this claim again in George Csicsery 's documentary film "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture" while discussing his difficulties at Purdue and in

168-643: A 2021 science fiction film " The Infinites ", a 1953 science fiction short story by Philip K. Dick The Infinites, a fictional group of cosmic beings in the Avengers Infinity comic book series Infinite, a character in the video game Sonic Forces Infinite Flight , a flight simulator released on 2011 Halo Infinite , 2021 video game Infinity symbol , ∞ See also [ edit ] Infinity (disambiguation) All pages with titles beginning with Infinite All pages with titles containing Infinite Topics referred to by

210-528: A New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop. A profile published in the Quanta Magazine reports that Zhang used to live in his car during the initial job-hunting days. He served as lecturer at UNH from 1999 until around January 2014, when UNH appointed him to a full professorship as a result of his breakthrough on prime numbers. Zhang stayed for

252-480: A given threshold n and the number of all primes less than n tends to 1 as n tends to infinity. Yitang Zhang Yitang Zhang ( Chinese : 张益唐 ; born February 5, 1955) is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015. Previously working at the University of New Hampshire as

294-546: A lecturer, Zhang submitted a paper to the Annals of Mathematics in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize , a 2014 Cole Prize , a 2014 Rolf Schock Prize , and a 2014 MacArthur Fellowship . Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015. Zhang

336-494: A multiple of 5. This is not prime unless n  = 1. Likewise, if n were to end in 4 or 9, 6 n would end in 4, and 6 n  +1 would be a multiple of 5. The same rule applies modulo any prime p ≥ 5: If n ≡ ±6 (mod p ), then one of the pair will be divisible by p and will not be a twin prime pair unless 6 n = p  ±1. p = 5 just happens to produce particularly simple patterns in base 10. An isolated prime (also known as single prime or non-twin prime )

378-518: A semester at the Institute for Advanced Study in Princeton , NJ, in 2014, and he joined the University of California, Santa Barbara in fall 2015. On April 17, 2013, Zhang announced a proof that there are infinitely many pairs of prime numbers that differ by less than 70 million. This result implies the existence of an infinitely repeatable prime 2-tuple , thus establishing a theorem akin to

420-399: Is a prime number p such that neither p  − 2 nor p  + 2 is prime. In other words, p is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both composite . The first few isolated primes are It follows from Brun's theorem that almost all primes are isolated in the sense that the ratio of the number of isolated primes less than

462-461: Is a recipient of the 2014 MacArthur award , and was elected as an Academia Sinica Fellow during the same year. He was an invited speaker at the 2014 International Congress of Mathematicians. In 1989 Zhang joined a group interested in Chinese democracy ( 中国民联 ). In a 2013 interview, he affirmed that his political views on the subject had not changed since. Infinite From Misplaced Pages,

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504-467: Is also 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 + 2 k is also 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 (see below), postulates a distribution law for twin primes akin to

546-414: Is conjectured to equal twice the twin prime constant ( OEIS :  A114907 ) (not to be confused with Brun's constant ), according to the Hardy–Littlewood conjecture. Every third odd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition

588-600: Is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P ( k ) is true. The classical form of the twin prime conjecture is equivalent to P (2); and in fact it has been conjectured that P ( k ) holds for all even integers k . While these stronger conjectures remain unproven, a result due to James Maynard in November 2013, employing a different technique, showed that P ( k ) holds for some k ≤ 600. Subsequently, in April 2014,

630-476: Is false. This conjecture has been extended by Dickson's conjecture . Polignac's conjecture from 1849 states that for every positive even integer k , there are infinitely many consecutive prime pairs p and p′ such that p ′ − p = k (i.e. there are infinitely many prime gaps of size k ). The case k = 2 is the twin prime conjecture . The conjecture has not yet been proven or disproven for any specific value of k , but Zhang's result proves that it

672-417: Is the twin prime constant (slightly less than 2/3), given below . The twin prime conjecture claims that the set of prime pairs is infinite . 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

714-776: Is true for at least one (currently unknown) value of k . Indeed, if such a k did not exist, then for any positive even natural number N there are at most finitely many n such that p n + 1 − p n = m {\displaystyle p_{n+1}-p_{n}=m} for all m < N and so for n large enough we have p n + 1 − p n > N , {\displaystyle p_{n+1}-p_{n}>N,} which would contradict Zhang's result. Beginning in 2007, two distributed computing projects, Twin Prime Search and PrimeGrid , have produced several record-largest twin primes. As of August 2022,

756-661: The Polymath project 8 lowered the bound to k ≤ 246. If the Elliott–Halberstam conjecture and its generalization, respectively, hold, then k ≤ 12 and k ≤ 6 follow using current methods. Zhang was awarded the 2013 Morningside Special Achievement Award in Mathematics , the 2013 Ostrowski Prize , the 2014 Frank Nelson Cole Prize in Number Theory, and the 2014 Rolf Schock Prize in Mathematics. He

798-423: The prime number theorem . On 17 April 2013, Yitang Zhang announced a proof that there exists an integer N that is less than 70 million, where there are infinitely many pairs of primes that differ by  N . Zhang's paper was accepted in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound. One year after Zhang's announcement,

840-534: The twin prime conjecture . Zhang's paper was accepted by Annals of Mathematics in early May 2013, his first publication since his last paper in 2001. The proof was refereed by leading experts in analytic number theory . Researchers built off of Zhang's result like in Polymath8 project . If P ( N ) stands for the proposition that there is an infinitude of pairs of prime numbers (not necessarily consecutive primes) that differ by exactly N , then Zhang's result

882-465: The Jacobian conjecture, "never published any paper on algebraic geometry" after leaving Purdue, and "wasted seven years of his own life and my time". After some years, Zhang managed to find a position as a lecturer at the University of New Hampshire , where he was hired by Kenneth Appel in 1999. Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for

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924-585: The Polymath Project and James Maynard, this bound has been substantially reduced, with the current best known value being 246. The first Hardy–Littlewood conjecture (named after G. H. Hardy and John Littlewood ) is a generalization of the twin prime conjecture. It is concerned with the distribution of prime constellations , including twin primes, in analogy to the prime number theorem . Let ⁠ π 2 ( x ) {\displaystyle \pi _{2}(x)} ⁠ denote

966-648: The President of Peking University, and Professor Deng Donggao, chair of the university's Math Department, Zhang was granted a full scholarship at Purdue University . Zhang arrived at Purdue in January 1985, studied there for six and a half years, and obtained his PhD in mathematics in December 1991. Zhang's PhD work was on the Jacobian conjecture . After graduation, Zhang had trouble finding an academic position. In

1008-407: The bound had been reduced to 246, where it remains. These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao . This second approach also gave bounds for the smallest f ( m ) needed to guarantee that infinitely many intervals of width f ( m ) contain at least m primes. Moreover (see also

1050-414: The bound to c = 0.085786... In 2005, Goldston, János Pintz , and Yıldırım made a breakthrough by proving that c could be chosen to be arbitrarily small, i.e. However, this result does not extend to slower-growing functions, such as c ln ln p , where the question remains open. Under the assumption of the Elliott–Halberstam conjecture (or a slightly weaker version), Goldston, Pintz, and Yıldırım proved

1092-583: The current largest twin prime pair known is 2996863034895 × 2 ± 1 , with 388,342 decimal digits. It was discovered in September ;2016. There are 808,675,888,577,436 twin prime pairs below 10. An empirical analysis of all prime pairs up to 4.35 × 10 shows that if the number of such pairs less than x is f  ( x ) · x  /(log x ) then f  ( x ) is about 1.7 for small x and decreases towards about 1.3 as x tends to infinity. The limiting value of f  ( x )

1134-508: The existence of a constant c < 1 such that there are infinitely many primes p where p' − p < c ln p (where p' denotes the next prime after p ). This result established that there exist infinitely many pairs of consecutive primes whose gap grows more slowly than a logarithmic function . Subsequent mathematicians improved upon Erdős's work: In 1986, Helmut Maier showed that the constant could be reduced to c < 0.25 . In 2004, Daniel Goldston and Cem Yıldırım further improved

1176-539: The existence of infinitely many integers n where at least two numbers from the set {n, n + 2, n + 6, n + 8, n + 12, n + 18, n + 20} are prime. With stronger hypotheses, they showed that infinitely often at least two numbers from the smaller set {n, n + 2, n + 4, n + 6} are prime. In 2013, Yitang Zhang proved that: where N = 7 × 10^7 . This represented a significant improvement over the Goldston–Graham–Pintz–Yıldırım result. Through subsequent work by

1218-583: The fields. He worked as a laborer for 10 years and was unable to attend high school. After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received a Bachelor of Science in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained a Master of Science in mathematics in 1984. After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun ,

1260-603: The 💕 [REDACTED] Look up infinite in Wiktionary, the free dictionary. Infinite may refer to: Mathematics [ edit ] Infinite set , a set that is not a finite set Infinity , an abstract concept describing something without any limit Music [ edit ] Performers [ edit ] Infinite (group) , a South Korean boy band Infinite (rapper) , Canadian rapper Albums [ edit ] Infinite (Deep Purple album) , 2017 Infinite (Eminem album) or

1302-431: The next section) assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath Project wiki states that the bound is 12 and 6, respectively. A strengthening of Goldbach’s conjecture , if proved, would also prove there is an infinite number of twin primes, as would the existence of Siegel zeroes . In 1940, Paul Erdős proved a significant result about gaps between consecutive primes. He demonstrated

Twin prime - Misplaced Pages Continue

1344-498: The number between the two primes is a multiple of 6. As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12. In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent . 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

1386-600: The number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C 2 as C 2 = ∏ p p r i m e , p ≥ 3 ( 1 − 1 ( p − 1 ) 2 ) ≈ 0.660161815846869573927812110014 … . {\displaystyle C_{2}=\prod _{\textstyle {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .} (Here

1428-599: The number of twin primes less than N does not exceed for some absolute constant C > 0. In fact, it is bounded above by 8 C 2 N ( log ⁡ N ) 2 [ 1 + O ⁡ ( log ⁡ log ⁡ N log ⁡ N ) ] , {\displaystyle {\frac {8C_{2}N}{(\log N)^{2}}}\left[1+\operatorname {\mathcal {O}} \left({\frac {\log \log N}{\log N}}\right)\right],} where C 2 {\displaystyle C_{2}}

1470-549: The pair (2, 3) is not considered to be a pair of twin primes. Since 2 is the only even prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes. The first several twin prime pairs are Five is the only prime that belongs to two pairs, as every twin prime pair greater than (3, 5) is of the form ( 6 n − 1 , 6 n + 1 ) {\displaystyle (6n-1,6n+1)} for some natural number n ; that is,

1512-414: The prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for ⁠ π 2 ( x ) {\displaystyle \pi _{2}(x)} ⁠ above. The fully general first Hardy–Littlewood conjecture on prime k -tuples (not given here) implies that the second Hardy–Littlewood conjecture

1554-557: The product extends over all prime numbers p ≥ 3 .) Then a special case of the first Hardy-Littlewood conjecture is that π 2 ( x ) ∼ 2 C 2 x ( ln ⁡ x ) 2 ∼ 2 C 2 ∫ 2 x d t ( ln ⁡ t ) 2 {\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}} in

1596-413: The same term [REDACTED] This disambiguation page lists articles associated with the title Infinite . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Infinite&oldid=1228614554 " Category : Disambiguation pages Hidden categories: Short description

1638-401: The sense that the quotient of the two expressions tends to 1 as x approaches infinity. (The second ~ is not part of the conjecture and is proven by integration by parts .) The conjecture can be justified (but not proven) by assuming that ⁠ 1 ln ⁡ t {\displaystyle {\tfrac {1}{\ln t}}} ⁠ describes the density function of

1680-519: The title song (see below), 1996 Infinite (Sam Concepcion album) , 2013 Infinite (Stratovarius album) , 2000 The Infinite (album) , by Dave Douglas, 2002 Infinite , by Kazumi Watanabe , 1971 Infinite , an EP by Haywyre , 2012 Songs [ edit ] "Infinite" (Beni Arashiro song) , 2004 "Infinite" (Eminem song) , 1996 "Infinite" (Notaker song) , 2016 "Infinite", by Forbidden from Twisted into Form , 1990 Other uses [ edit ] Infinite (film) ,

1722-632: The years that followed. Moh claimed that Zhang never came back to him requesting recommendation letters. In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh's support, and, having published no papers, was unable to find an academic job". In 2018, responding to reports of his treatment of Zhang, Moh posted an update on his website. Moh wrote that Zhang "failed miserably" in proving

Twin prime - Misplaced Pages Continue

1764-653: Was born in Shanghai, China, with his ancestral home in Pinghu , Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University . At around the age of nine, he found a proof of the Pythagorean theorem . He first learned about Fermat's Last Theorem and Goldbach's conjecture when he was 10. During the Cultural Revolution , he and his mother were sent to the countryside to work in

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