When teaching Euclid's classic proof of the infinitude of primes today, the following question appeared to me. Let $p_1,p_2,p_3,\ldots$ be the prime numbers, listed in increasing order. Set

$$k_n = p_1 p_2 \cdots p_n + 1.$$

The numbers $k_n$ do not need to be prime, but they seem to be remarkably often. In particular, the first non-prime is

$$k_6 = (2)(3)(5)(7)(11)(13)+1 = 30031 = (59)(509).$$

Question: are infinitely many of the $k_n$ prime? My guess is that this is wide open; if it is, what do the usual heuristics for primes tell us we should expect?