Heuristically, the probability of such a chain starting from a given prime p is of the form C_1 * C_2 * C_3 ... where C_k —> 0 as k —> oo. If we then *add* these probabilities over all primes p, we get an infinitesimal that is countable * 1/continuum that is heuristically *greater than* the chance of an infinite Cunningham chain, suggesting it's pretty unlikely. —Dan W. Edwin Clark wrote: ----- This matter seems to be related to Cunningham chains <https://en.wikipedia.org/wiki/Cunningham_chain>--sequences of primes p_i such that p_{i+1} = 2p_{i} + 1. ... ... It is conjectured that there are Cunningham chains of length k for all positive integers k. Is there a reason that one could not have a Cunningham chain of infinite length? -----