Thanks to the machinery underlying the Green-Tao theorem a lot of numerical observations on primes in arithmetic progressions now have a chance to be formulated precisely, and possibly proved one day.
Here is one such observation (I’m not claiming it to be new or interesting, I haven’t actually tried to even browse the literature nor the web).
Take the primes:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53…
and extract the longest progression from where you stand (starting at 3, since 2 is of different parity) , one obtains:
3 5 7
11 17 23 29
13 37 61
19 31 43
41 47 53 59
67 73 79
71 89 107
83 131 179 227
97 103 109
101 137 173
113 191 269 347
127 139 151 163
149 233 317 401
157 193 229
167 239 311 383
181 211 241 271
197 293 389
199 283 367
223 277 331
251 257 263
281 419 557
307 373 439
313 457 601
337 379 421 463
349 541 733
353 431 509 587
359 461 563
397 487 577
409 619 829 1039 1249 1459 1669 1879 2089
433 523 613
443 467 491
I have data for the first 10,000 primes from which the average length of those APs is 3.2, so there’s clearly an abundance of 3-term ones, with very occasional longer ones.
Question: do we nevertheless get arbitrarily long APs when partitioning the primes this way?
It doesn’t seem so, which begs the question: is there a way to partition the primes into 3-term APs only? If there is, what would this say about the set of primes? (that it is somehow 3-periodic maybe?) I’ll ask this over at MathOverflow…