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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] |
Nikolai Chudakov
Nikolai Grigor'evich Chudakov (russian: Никола́й Григо́рьевич Чудако́в; 1904–1986) was a Russian and Soviet mathematician. He was born in Lysovsk, Novo-Burassk, Saratov, Russian Empire. His father worked as a medical assistant. Biography He first studied at the Faculty of Physics and Mathematics at Saratov State University, but then he transferred to Moscow University. He then graduated in 1927. In 1930, he was named head of higher mathematics at Saratov University. In 1936, he successfully defended his thesis and became a Doctor of Science. Among others, he considerably improved a result from Guido Hoheisel and Hans Heilbronn on an upper bound for prime gaps. He worked in Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ... until 1940, but then ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yitang Zhang
Yitang Zhang (; 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 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 award. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015. Early life and education Zhang 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 the Goldbach conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cem Yıldırım
Cem Yalçın Yıldırım (born 8 July 1961) is a Turkish mathematician who specializes in number theory. He obtained his B.Sc from Middle East Technical University in Ankara, Turkey and his PhD from the University of Toronto in 1990. His advisor was John Friedlander. He is currently a faculty member at Boğaziçi University in Istanbul, Turkey. In 2005(), with Dan Goldston and János Pintz, he proved, that for any positive number ''ε'' there exist primes ''p'' and ''p''′ such that the difference between ''p'' and ''p''′ is smaller than ''ε'' log ''p''. Formally; :\liminf_\frac=0 where ''p''''n'' denotes the ''n''th prime number. In other words, for every ''c'' > 0, there exist infinitely many pairs of consecutive primes ''p''''n'' and ''p''''n''+1 which are closer to each other than the average distance between consecutive primes by a factor of ''c'', i.e., ''p''''n''+1 − ''p''''n'' < ''c'' log ''p''''n''. This r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Goldston
Daniel Alan Goldston (born January 4, 1954 in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University. Early life and education Daniel Alan Goldston was born on January 4, 1954 in Oakland, California. In 1972, he matriculated to the University of California, Berkeley, where he earned his bachelor's degree and, in 1981, a Ph.D. in mathematics. His doctoral advisor at Berkeley was Russell Sherman Lehman; his dissertation was entitled "Large Differences between Consecutive Prime Numbers". Career After earning his doctorate, Goldston worked at the University of Minnesota Duluth and then spent the next academic year (1982–83) at the Institute for Advanced Study (IAS) in Princeton. He has worked at San Jose State University since 1983, save for stints at the IAS (1990), the University of Toronto (1994), and the Mathematical Sciences Research Institute in Berkeley (1999). Research Go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Pintz
János Pintz (born 20 December 1950 in Budapest) is a Hungary, Hungarian mathematician working in analytic number theory. He is a fellow of the Alfréd Rényi Institute of Mathematics, Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences. In 2014, he received the Cole Prize. Mathematical results Pintz is best known for proving in 2005 (with Daniel Goldston and Cem Yıldırım) that :: \liminf_\frac=0 where p_n denotes the ''n''th prime number. In other words, for every ε > 0, there exist infinitely many pairs of consecutive primes ''p''''n'' and ''p''''n''+1 that are closer to each other than the average distance between consecutive primes by a factor of ε, i.e., ''p''''n''+1 − ''p''''n'' < ε log ''p''''n''. This result was originally reported in 2003 by Daniel Goldston and Cem Yıldırım but was later retracted. Pintz joined the team and comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glyn Harman
Glyn Harman (born 2 November 1956) is a British mathematician working in analytic number theory. One of his major interests is prime number theory. He is best known for results on gaps between primes and the greatest prime factor of ''p'' + ''a'', as well as his lower bound for the number of Carmichael numbers up to X. His monograph ''Prime-detecting Sieves'' (2007) was published by Princeton University Press. He has also written a book ''Metric Number Theory'' (1998). As well, he has contributed to the field of Diophantine approximation. Harman also proved that there are infinitely many primes (additive primes) whose sum of digits is prime. (the sequencA046704in the OEIS).The OEIadditive primes/ref> Harman retired at the end of 2013 from being a professor at Royal Holloway, University of London. Previously he was a professor at Cardiff University. Harman is married, and has three sons, and used to live in Wokingham, Berkshire before moving to Harrow, Middlesex/Greater London ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Huxley
Martin Neil Huxley (born in 1944) is a British mathematician, working in the field of analytic number theory. He was awarded a PhD from the University of Cambridge in 1970, the year after his supervisor Harold Davenport had died. He is a professor at Cardiff University. Huxley proved a result on gaps between prime numbers, namely that if ''p''''n'' denotes the ''n''-th prime number and if θ > 7/12, then : p_ - p_n < p_n^\theta, for all ''n''. Huxley also improved the known bound on the Dirichlet divisor problem
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number ...
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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] |
Lindelöf Hypothesis
In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It says that for any ''ε'' > 0, \zeta\!\left(\frac + it\right)\! = O(t^\varepsilon) as ''t'' tends to infinity (see big O notation). Since ''ε'' can be replaced by a smaller value, the conjecture can be restated as follows: for any positive ''ε'', \zeta\!\left(\frac + it\right)\! = o(t^\varepsilon). The μ function If σ is real, then ''μ''(σ) is defined to be the infimum of all real numbers ''a'' such that ζ(σ + ''iT'' ) = O(''T'' ''a''). It is trivial to check that ''μ''(σ) = 0 for σ > 1, and the functional equation of the zeta function implies that ''μ''(σ) = ''μ''(1 − σ) − σ + 1/2. The Phragmén–Lindelöf theorem implies that ''μ'' is a conve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately : \frac x where log is the natural logarithm, in the sense that :\lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is :\lim_\pi(x) / \operatorname(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
