Let p=3 \cdot 10^{10}+1 be a prime and let p_{n} denote the probability that p \mid\left(k^{k}-1\right) for a random k chosen uniformly from \{1,2, \cdots, n\}. Given that p_{n} \cdot p converges to a value L as n goes to infinity, what is L?
Let p=3 \cdot 10^{10}+1 be a prime and let p_{n} denote the probability that p \mid\left(k^{k}-1\right) for a random k chosen uniformly from \{1,2, \cdots, n\}. Given that p_{n} \cdot p converges to a value L as n goes to infinity, what is L?