Consecutive primes in arithmetic progression

This article needs more work.

Please help by expanding it!

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.

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).
• In 1967, Lander and Parkin found six consecutive primes in arithmetic progression: (121174811 + 30k, k = 0, ..., 5).
• In 1995, Dubner and Nelson found seven consecutive primes in arithmetic progression: ?.
• In 1997, Dubner, Forbes, Lygeros, Mizony, Nelson and Zimmermann found eight consecutive primes in arithmetic progression: ?.
• In 1998, Toplic found nine consecutive primes in arithmetic progression: ?.
• In 1998, Toplic found ten consecutive primes in arithmetic progression, A033290.

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);
• a (6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);
• a (7) = ?: (?, ? + k, ? + 2k, ? + 3k, ? + 4k, ? + 5k, ? + 6k), k ≥ 210;
• 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.

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

• Primes in arithmetic progression
• OEIS Index: sequences related to primes in arithmetic progression

External links

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