Primes in arithmetic progression

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In 2004, Ben Green and Terrence Tao published a preprint proving that there exist sequences of (not necessarily consecutive!) primes in arithmetic progression, such as A033168, of any length.

Sequences

A005115 Let

i, i + d, i + 2 d, ..., i + (n  −  1)  d

be an

n

-term arithmetic progression of primes; choose the one which minimizes the last term

a (n) = i + (n  −  1)  d

.

{2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, ...}

A113827 Initial terms associated with the arithmetic progressions of primes in A005115.

{2, 2, 3, 5, 5, 7, 7, 199, 199, 199, 110437, 110437, 4943, 31385539, 115453391, 53297929, 3430751869, 4808316343, 8297644387, 214861583621, 5749146449311, ...}

A093364 Gaps associated with the arithmetic progressions of primes in A005115.

{0, 1, 2, 6, 6, 30, 150, 210, 210, 210, 13860, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890, ...}

Number of primes in AP with given gaps and starting point

The following sequences give the maximum length, i.e., number of primes in AP with a given gap and starting point:

A088420 (Maximum) number of primes in arithmetic progression starting with

i = 3

and with

d = 2 n

.

{3, 3, 1, 3, 3, 1, 3, 2, 1, 3, 1, 1, 2, 3, 1, 1, 3, 1, 3, 3, 1, 2, 1, 1, 3, 1, 1, 2, 2, 1, 1, 3, 1, 3, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 3, 2, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, ...}

A088421 (Maximum) number of primes in arithmetic progression starting with

i = 5

and with

d = 2 n

.

{2, 1, 5, 2, 1, 5, 2, 1, 4, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 5, 1, 1, 5, 1, 1, 4, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 5, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 4, ...}

A088422 (Maximum) number of primes in arithmetic progression starting with

i = 7

and with

d = 2 n

.

{1, 2, 3, 1, 2, 4, 1, 2, 1, 1, 2, 2, 1, 1, 6, 1, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 3, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, ...}

A088423 (Maximum) number of primes in arithmetic progression starting with

i = 11

and with

d = 2 n

.

{2, 1, 4, 2, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 5, 2, 1, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 6, 2, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, ...}

A088424 (Maximum) number of primes in arithmetic progression starting with

i = 13

and with

d = 2 n

.

{1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 4, 1, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 6, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 1, ...}

A088425 (Maximum) number of primes in arithmetic progression starting with

i = 17

and with

d = 2 n

.

{2, 1, 3, 1, 1, 4, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 1, 2, ...}

A088426 (Maximum) number of primes in arithmetic progression starting with

i = 19

and with

d = 2 n

.

{1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 4, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, ...}

A088427 (Maximum) number of primes in arithmetic progression starting with

i = 23

and with

d = 2 n

.

{1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 1, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 4, 2, 1, 3, 2, 1, 2, 1, 1, 1, ...}

A088428 (Maximum) number of primes in arithmetic progression starting with

i = 29

and with

d = 2 n

.

{2, 1, 1, 2, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 4, 1, 1, 3, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1, 4, 1, 1, 1, 1, 1, 1, ...}

A088429 (Maximum) number of primes in arithmetic progression starting with

i = 31

and with

d = 2 n

.

{1, 1, 3, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, ...}

Other sequences concerning primes in AP

A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

{199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089}

This 10-tuple is row 10 of tables A086786, A113470, A133276, A133277.  ^{(add names of these sequences)} 

See A094220 for other initial terms of 10 primes in AP with

d = 210

.

(...)

Consecutive primes in arithmetic progression (CPAP)

See also consecutive primes in arithmetic progression for many sequences concerning these more restrictive cases, in particular CPAP with given gap.

See also

• Index entries for sequences related to primes in arithmetic progressions
• Consecutive primes in arithmetic progression
• Szemerédi’s theorem

External links

• Szemerédi's theorem—Wikipedia.org.