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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: (10 10 + 24493 + 30k, k = 0, ..., 4).[2]
- In 1967, Lander and Parkin found six consecutive primes in arithmetic progression: (121174811 + 30k, k = 0, ..., 5).[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 + 6k, 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 (see
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]
CPAP with given gap
Several sequences in OEIS deal with CPAP with given common differences. In the table below,
is the length of the multiplet (as in CPAP-
):
gap \ n: 2 3 4 5 6
-------------------------------------------------------------------------------------------------------------
2 | A001359 {3} (twin primes for n = 2, only {3} for n = 3, and inexistant for n > 3)
4 | A023200 (cousin primes, inexistant for n > 2)
6 | A031924 A047948 A033451
12 | A031930 A052188 A033447
18 | A031936 A052189 A033448
24 | A098974 A052190 A052242
30 | A124596 A052195 A052243 (A059044) (A058362) <- for n >= 5: arbitrary gap, but initial terms have 30
36 | A134117 A052197 A058252
42 | A134120 A052198 A058323
48 | A134123 A067388
60 | A126771 A089234 A210683 A210727
any | A054800 A059044 A058362
Sequence
A052239 lists the start of the first CPAP-
4 with common gap
.
See also
References
- ↑ Chris K. Caldwell, Consecutive Primes in Arithmetic Progression.
- ↑ M. F. Jones, M. Lal and W. J. Blundon (1967). “Statistics on certain large primes”. Math. Comp. 21 (97): pp. 103–107. MR 36:3707
- ↑ 3.0 3.1 3.2 L. J. Lander and T. R. Parkin (1967). “Consecutive primes in arithmetic progression”. Math. Comp. (AMS) 21: p. 489. http://www.ams.org/mcom/1967-21-099/S0025-5718-67-99147-8/S0025-5718-67-99147-8.pdf.
- ↑ 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. (AMS) 71 (239): pp. 1323–1328. http://dx.doi.org/10.1090/S0025-5718-01-01374-6.
- ↑ Jens Kruse Andersen (2014). “The minimal CPAP-k ”. http://primerecords.dk/cpap.htm#minimal. “Heuristics (estimates based on probability) indicate the minimal CPAP-7 may have 22 or 23 digits. The smallest known is 32 digits: 19252884016114523644357039386451 + 210 n, n = 0..6.”
- ↑ Weisstein, Eric W., Primorial, from MathWorld—A Wolfram Web Resource.
External links