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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]}

## Contents

## 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

- ↑ Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
- ↑ M. F. Jones, M. Lal and W. J. Blundon, "Statistics on certain large primes," Math. Comp., 21:97 (1967) 103--107. MR 36:3707
- ↑
^{3.0}^{3.1}^{3.2}L. J. Lander and T. R. Parkin, "Consecutive primes in arithmetic progression," Math. Comp., 21 (1967) 489 - ↑
^{4.0}^{4.1}Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann, Background (archived by Manfred Toplic) - ↑
^{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 . - ↑ 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"
- ↑ Eric W. Weisstein. "Primorial".
*MathWorld*. wolfram.com .

## External links

- Jens Kruse Andersen, The Largest Known CPAP's, 2014.
- Manfred Toplic, The nine and ten primes project, 2004.