Affine Prime Chains and a Conjecture on Arbitrarily Long Cunningham Chains

Affine Prime Chains and a Conjecture on Arbitrarily Long Cunningham Chains This preprint presents a new conjecture proposed by Kamyl Ababsa, a 15-year-old independent researcher, about the existence of arbitrarily long sequences of prime numbers generated by the recurrence pn+1=2pn+1p_{n+1} = 2p_n + 1pn+1=2pn+1. The main statement — the Affine Prime Chain Conjecture — asserts that for every integer kkk, there exist infinitely many primes ppp such that the first kkk terms of the sequence p,2p+1,4p+3,8p+7,…p, 2p+1, 4p+3, 8p+7, \dotsp,2p+1,4p+3,8p+7,… are all prime. The note also discusses local admissibility, Bateman–Horn heuristics, numerical evidence, and an open question about the possibility of infinite prime chains. [email protected] I would also like to mention a friend of mine, Lyna Mostefa, who contributed to the progress of my work, although she was not directly involved in the mathematical aspects.