Arithmetic Progressions From Prime Numbers
Szemerédi's theorem states that a set of natural numbers of non-zero upper asymptotic density contains finite arithmetic progressions, of any arbitrary length k.
Erdős made a more general conjecture from which it would follow that
- 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.
See also Dirichlet's theorem on arithmetic progressions.
As of 2010, the longest known arithmetic progression of primes has length 26:
- 43142746595714191 + 23681770·23#·n, for n = 0 to 25. (23# = 223092870)
As of 2011, the longest known arithmetic progression of consecutive primes has length 10. It was found in 1998. The progression starts with a 93-digit number
- 100 99697 24697 14247 63778 66555 87969 84032 95093 24689
- 19004 18036 03417 75890 43417 03348 88215 90672 29719
and has the common difference 210.
Read more about this topic: Problems Involving Arithmetic Progressions
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