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Consecutive primes in arithmetic progression

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It is conjectured that there are arithmetic progressions of consecutive primes for any , but this has not even been (logically) proved for , although there are proofs by examples for up to 10. It seems likely that there are infinitely many CPAP- with prime difference , for all and , where is the primorial of and CPAP stands for Consecutive Primes in Arithmetic Progression.[1]

Occurrences of n CPAP

  • In 1967, Jones, Lal and Blundon found five consecutive primes in arithmetic progression, A293791: ().[2]
  • In 1967, Lander and Parkin found six consecutive primes in arithmetic progression: ().[3]
  • In 1995, Dubner and Nelson found seven consecutive primes in arithmetic progression:[4]
  • In 1997, Dubner, Forbes, Lygeros, Mizony, Nelson and Zimmermann found eight consecutive primes in arithmetic progression:[4]
  • In 1998, Toplic found nine consecutive primes in arithmetic progression:[5]
  • In 1998, Toplic found ten consecutive primes in arithmetic progression, A033290.[5]

First and smallest occurrence of n CPAP

The first and smallest occurrence of consecutive primes in arithmetic progression are listed in A006560:

  • a(1) = 2: (2) (degenerate arithmetic progression);
  • a(2) = 2: (2, 3);
  • a(3) = 3: (3, 5, 7);
  • a(4) = 251: (251, 257, 263, 269); (not 5+k*6, because it skips 7,13,19)
  • a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);[3]
  • a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);[3]
  • a(7) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k), k ≥ 210;[6]
  • a(8) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k), k ≥ 210; The expected size is a(8) > ?.
  • a(9) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k, ? + 8k), k ≥ 210; The expected size is a(9) > ?.
  • a(10) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k, ? + 7k, ? + 8k, ? + 9k), k ≥ 210; The expected size is a(10) > ?.

Common differences of first and smallest occurrence of n CPAP

The common differences of first and smallest arithmetic progression of consecutive primes are (Cf. A126989)

{0, 1, 2, 6, 30, 30, ≥ 210, ≥ 210, ≥ 210, ≥ 210, ≥ 2310, ≥ 2310, ≥ 30030, ...}
or {0, 1#, 2#, 3#, 4#, 4#, ≥ 5#, ≥ 5#, ≥ 5#, ≥ 5#, ≥ 6#, ≥ 6#, ≥ 7#, ...}

where a(7)=210 is unconfirmed and is the th primorial listed in A002110.[7]

See also

References

  1. Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
  2. M. F. Jones, M. Lal and W. J. Blundon, "Statistics on certain large primes," Math. Comp., 21:97 (1967) 103--107. MR 36:3707
  3. 3.0 3.1 3.2 L. J. Lander and T. R. Parkin, "Consecutive primes in arithmetic progression," Math. Comp., 21 (1967) 489
  4. 4.0 4.1 Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann, Background (archived by Manfred Toplic)
  5. 5.0 5.1 Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann (2001). "Ten Consecutive Primes in Arithmetic Progression". Math. Comp. Volume 71, Number 239. AMS. Pages 1323–1328. 
  6. Jens Kruse Andersen (2014). "The minimal CPAP-k". "Heuristics (estimates based on probability) indicate the minimal CPAP-7 may have 22 or 23 digits. The smallest known is 32 digits: 19252884016114523644357039386451 + 210n, n=0..6" 
  7. Eric W. Weisstein. "Primorial". MathWorld. wolfram.com. 

External links