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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.
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.
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.
Certain topics have not yet reached book form in any depth. Some examples are (i) Montgomery's pair correlation conjecture and the work that initiated from it, (ii) the new results of Goldston, Pintz and Yilidrim on small gaps between primes, and (iii) the Green–Tao theorem showing that arbitrarily long arithmetic progressions of primes exist.
In particular, jointly with Terence Tao, they proved a structure theorem [8] for approximate groups, generalising the Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has worked, jointly with Kevin Ford and Sean Eberhard , on the theory of the symmetric group , in particular on what proportion of its elements fix a set ...
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This work was further developed by Ben Green and Terence Tao, leading to the Green–Tao theorem. In 2003, Gowers established a regularity lemma for hypergraphs, [15] analogous to the Szemerédi regularity lemma for graphs. In 2005, he introduced [16] the notion of a quasirandom group.