Fibonacci primes, primes of the form 2n − k and beyond

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences ${\left(u_n\right)}_{n\ge 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes $u_n$, or else there exists a constant $c_u>0$ (which we can give good approximations to) such that there are $\sim c_u\log N$ primes $u_n$ with $n\le N$, as $N\to \infty$. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the $u_n\left(\mathrm{mod}p\right)$.