Green-Tao theorem
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In mathematics, the Green-Tao theorem, proved by Ben Green and Terence Tao in 2004[1], states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for any natural number k, there exist k-term arithmetic progressions of primes.
In 2006, Tao and Tamar Ziegler extended the result to cover polynomial progressions.[2] More precisely, given any integer-valued polynomials P1,..., Pk in one unknown m with vanishing constant terms, there are infinitely many integers x, m such that x + P1(m), ..., x + Pk(m) are simultaneously prime. The special case when the polynomials are m, 2m, ..., km implies the previous result that there are length k arithmetic progressions of primes.
In other words, given a number k, there exists an arithmetic progression, of primes, of length k and the spacing between each of these primes is m, where m exists but may not be known.
Since they are existence theorems, they do not show how to find the progressions. On January 18, 2007, Jaroslaw Wroblewski found the first known case of 24 primes in arithmetic progression:[3]
- 468395662504823 + 205619 × 23# × n, for n = 0 to 23 (23# = 223092870)
[edit] See also
[edit] References
- ^ Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions,8 Apr 2004.
- ^ Terence Tao, Tamar Ziegler, The primes contain arbitrarily long polynomial progressions
- ^ Jens Kruse Andersen, Primes in Arithmetic Progression Records
