A Study of Cunningham Bounds through Rogue Primes

If $p$ is prime, a sequence of prime numbers $\{p,2p+1,4p+3,\mathrm{.}\mathrm{.}\mathrm{.},2^{n-1}(p+1)-1\}$ is called a Cunningham chain. These are finite sequences of prime numbers, for which each element but the last is a Sophie Germain prime. It is conjectured that there are arbitrarily large such Cunningham chains, and these chains form an essential part of the study of Sophie Germain primes. In this paper, we aim to significantly improve existing bounds for the length of Cunningham chains by considering their behaviour in the framework of what we will define as $\mathit{\text{rogueness}}$.