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Consecutive primes in arithmetic progression

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It is conjectured that there are arithmetic progressions of
n
consecutive primes
for any
n
, but this has not even been (logically) proved for
n = 3
, although there are proofs by examples for
n
up to 10. It seems likely that there are infinitely many CPAP-
n
with prime difference
c ⋅  n #
, for all
c
and
n
, where
n #
is the primorial of
n
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
n, n   ≥   1,
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
n, n   ≥   1,
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
n #
is the
n
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,
n
is the length of the multiplet (as in CPAP-
n
):
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
6n
.

See also

References

  1. Chris K. Caldwell, Consecutive Primes in Arithmetic Progression.
  2. 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. 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. 
  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. (AMS) 71 (239): pp. 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 + 210 n, n = 0..6. 
  7. Weisstein, Eric W., Primorial, from MathWorld—A Wolfram Web Resource.

External links