A Cunningham chain of length $l$ is a sequence of prime numbers $p_i$ such that $p_{i+1} = 2p_i+1$. This website lists the smallest prime that starts a complete Cunningham chain of length $l$, for $1<l<13$.
In it, they state that the smallest prime for a Cunningham chain of length $4$ is $509$. I disagree with this, since I can find a chain of length $4$ starting at a lower prime: $$ \{5, 11, 23, 47\}. $$
Therefore $5$ should be the lowest prime starting a Cunningham chain of length $4$. What am I missing? Or is the website wrong?
Your chain is not complete. You can include a $2$ at the beginning to make a chain of length $5$. The site isn't claiming that $509$ is the smallest prime that starts a chain of length $4$, they are claiming that it is the smallest prime that starts a complete chain of length $4$.