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Ben Joseph Green FRS (born 27 February 1977) is a British mathematician, ... In particular, jointly with Terence Tao, they proved a structure theorem [8] ...
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k {\displaystyle k} , there exist arithmetic progressions of primes with k {\displaystyle k} terms.
Any given arithmetic progression of primes has a finite length. In 2004, Ben J. Green and Terence Tao settled an old conjecture by proving the Green–Tao theorem: The primes contain arbitrarily long arithmetic progressions. [1] It follows immediately that there are infinitely many AP-k for any k.
Terence Chi-Shen Tao FAA FRS (Chinese: 陶哲軒; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences.
The Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, [3] states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, there exist arithmetic progressions of primes, with k terms, where k can be any natural number. The proof is an extension of Szemerédi's theorem.
This result was proven by Ben Green and Terence Tao in 2004 and is now known as the Green–Tao theorem. [3] See also Dirichlet's theorem on arithmetic progressions. As of 2020, the longest known arithmetic progression of primes has length 27: [4] 224584605939537911 + 81292139·23#·n, for n = 0 to 26. (23# = 223092870)
“When I called Ben, I said, ‘I hope you don’t have any allergies, because there’s every imaginable animal running around here,’” David Gordon Green, the film’s director, says.
A family of conjectures () was made by Ben Green and Terence Tao, concerning the Möbius function of prime number theory and -step nilsequences. Here the underlying Lie group G {\displaystyle G} is assumed simply connected and nilpotent with length at most s {\displaystyle s} .