# Essays/Prime APVs

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 exists a k-term APV of primes. Here we list prime APVs for k from 1 to 26, 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 ${\displaystyle \leftrightarrow }$  */ 2 3 5 7 ${\displaystyle \leftrightarrow }$  */ p: i.4 ${\displaystyle \leftrightarrow }$  */p:i.(p:^:_1) 1+7 ${\displaystyle \leftrightarrow }$  */@(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 + (pm 3) * i.4 5 11 17 23 5 5 + (pm 3) * i.5 5 11 17 23 29 6 7 + (pm 5) * i.6 7 37 67 97 127 157 7 7 + (3 * pm 5) * 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 26 43142746595714191x + (23681770x * pm 23) * i.26 43142746595714191 ... 175223597495211691

## 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
)
```