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The sequence of primes numbers contains arithmetic progressions of any length. 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]
Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...
Because the sum of the reciprocals of the primes diverges, the Green–Tao theorem on arithmetic progressions is a special case of the conjecture. The weaker claim that A must contain infinitely many arithmetic progressions of length 3 is a consequence of an improved bound in Roth's theorem. A 2016 paper by Bloom [4] proved that if {,..
Linnik's theorem (1944) concerns the size of the smallest prime in a given arithmetic progression. Linnik proved that the progression a + nd (as n ranges through the positive integers) contains a prime of magnitude at most cd L for absolute constants c and L. Subsequent researchers have reduced L to 5.
The problem of obtaining bounds in the k=3 case of Szemerédi's theorem in the vector space is known as the cap set problem. The Green–Tao theorem asserts the prime numbers contain arbitrarily long arithmetic progressions. It is not implied by Szemerédi's theorem because the primes have density 0 in the natural numbers.
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
There has been separate computational work to find large arithmetic progressions in the primes. The Green–Tao paper states 'At the time of writing the longest known arithmetic progression of primes is of length 23, and was found in 2004 by Markus Frind, Paul Underwood, and Paul Jobling: 56211383760397 + 44546738095860 · k ; k = 0, 1 ...
One major area of study in additive combinatorics are inverse problems: given the size of the sumset A + B is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions.