Is there any way (general procedure, i mean) to determine if a Euclid Number (En = pn# + 1) is prime or composite? Any research papers exploring this theme are also welcome. Thanks!

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compositenesis never very easy to establish (as one will never find a very small factor). – quid Aug 16 '13 at 11:28solarge that you can't actually compute with it; if it's the $n$-th "Euclid number" for $n$ on the order of $10^6$ then the only hope is to get lucky and find a prime factor $l$ small enough that $\prod_{m\leq n} p_n \equiv -1 \bmod l$. (If you want to find such an example, start with $l=p_r$ and try all $n < r$, which should succeed about $1-\exp(-1)$ of the time.) – Noam D. Elkies Aug 16 '13 at 13:09