Question 94
For a positive integer $a$, define a sequence of integers $x_1, x_2, \dots$ by letting $x_1 = a$ and $x_{n+1} = 2x_n + 1$ for $n \ge 1$. Let $y_n = 2^{x_n} - 1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_1, \dots, y_k$ are all prime.
(This problem was suggested by Valery Senderov from Russia.)
(This problem was suggested by Valery Senderov from Russia.)