An arithmetic progression vector is a vector of the form a + m * i.k for nonzero m . The Green-Tao theorem states that for any positive integer k there is exists a k-term APV of primes. Here we list prime APVs for k from 1 to 25, preferring APVs with small averages.
The expressions use the primorial function pm where pm n is the product of all the primes less than or equal to n . ("Primorial" is a combination of "prime" and "factorial".) For example, pm 7 
*/ 2 3 5 7 */ p: i.4 */p:i.(p:^:_1) 1+7 */@(i.&.(p:^:_1))@>: 7 .
k |
a + m * i.k |
APV |
1 |
2 + 0 * i.1 |
2 |
2 |
2 + 1 * i.2 |
2 3 |
3 |
3 + 2 * i.3 |
3 5 7 |
4 |
5 + 6 * i.4 |
5 11 17 23 |
5 |
5 + 6 * i.5 |
5 11 17 23 29 |
6 |
7 + 30 * i.6 |
7 37 67 97 127 157 |
7 |
7 + 150 * i.7 |
7 157 307 457 607 757 907 |
8 |
199 + (pm 7) * i.8 |
199 409 619 829 1039 1249 1459 1669 |
9 |
199 + (pm 7) * i.9 |
199 409 619 829 1039 1249 1459 1669 1879 |
10 |
199 + (pm 7) * i.10 |
199 409 619 829 1039 1249 1459 1669 1879 2089 |
11 |
110437 + (6 * pm 11) * i.11 |
110437 124297 138157 152017 165877 ... 249037 |
12 |
110437 + (6 * pm 11) * i.12 |
110437 124297 138157 152017 165877 ... 262897 |
13 |
4943 + (2 * pm 13) * i.13 |
4943 65003 125063 185123 245183 ... 725663 |
14 |
31385539 + (14 * pm 13) * i.14 |
31385539 31805959 32226379 32646799 ... 36850999 |
15 |
115453391 + (138 * pm 13) * i.15 |
115453391 119597531 123741671 ... 173471351 |
16 |
53297929 + (pm 19) * i.16 |
53297929 62997619 72697309 82396999 ... 198793279 |
17 |
3430751869x + (9x * pm 19) * i.17 |
3430751869x 3518049079x ... 4827507229x |
18 |
4808316343x + (74x * pm 19) * i.18 |
4808316343x 5526093403x ... 17010526363x |
19 |
8297644387x + (431x * pm 19) * i.19 |
8297644387x 12478210777x ... 83547839407x |
20 |
214861583621x + (1943x * pm 19) * i.20 |
214861583621x 233708081291x ... 572945039351x |
21 |
5749146449311x + (2681x * pm 19) * i.21 |
5749146449311x 5775151318201x ... 6269243827111x |
22 |
56211383760397x + (199678x * pm 23) * i.22 |
56211383760397 ... 991692883773457x |
23 |
56211383760397x + (199678x * pm 23) * i.23 |
56211383760397 ... 1036239621869317x |
24 |
468395662504823x + (205619x * pm 23) * i.24 |
468395662504823 ... 1523454717745013x |
25 |
6171054912832631x + (366384x * pm 23) * i.25 |
6171054912832631 ... 8132758706802551x |
Utilities
pm =: */ @ (i.&.(p:^:_1)) @ >: NB. primorial
papv=: 3 : 0 NB. check that y is a prime apv
assert. 1 p: y
assert. y = ({.y) + (i.#y) * (<:#y) %~ ({:y) - {.y
1
)
See also
Contributed by RogerHui.