Some arithmetic progressions of primes

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

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