Puzzle 1250 Iteration GPD(3*P+2)

Puzzle 1250 Iteration GPD(3*P+2)

On Dec 8, 2025 Alain Rochelli sent the following nice puzzle, as a follow-up to his Puzzle 1249.

Now, the 3*P+2 problem is as follows: start with any prime P (>=3) and consider the greatest prime divisor of 3*P+2, denoted A(P) = GPD(3*P+2).
Then, repeat the operation: A(A(P)), A(A(A(P))) … enough to fall into a cycle.

After computations carried out with the first primes up to 10^8, it is conjectured that all primes (>= 3) fall into a cycle included 5 or a cycle included 167 :

a) Cycle "5": 5, 17, 53, 23, 71, 43, 131, 79, 239, 719, 127, 383, 1151, 691, 83, 251, 151, 13, 41, 5 (19 terms)
b) Cycle "167": 167, 503, 1511, 907, 389, 167 (5 terms)

Q1. Are there more cycles other than the two mentioned above (5 & 167)?

Q2. Do you devise a way to produce a formal proof that there are only these two cycles, or this purpose is imposible to get (BTW this question Q2 is valid also for the Puzzle 1249)?