Primality of Euclid Numbers

Open Question
Consider the Euclid numbers:

Is there an infinite number of:
 * prime numbers
 * composite numbers

in this list?

It is known that $E_1$ to $E_5$ are all prime, and so is $E_{11}$.

The next such prime is $E_{75}$.

The sequence of indices of the Euclid primes begins:
 * $0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, \ldots$

However, whether the Euclid primes go on for ever is unknown.

Neither is it known whether the non-prime Euclid numbers is infinite.