TITLE: A question about numbers from Euclid's proof of infinitude of primes
QUESTION [13 upvotes]: Observe this list:
$$
\begin{aligned}
2+1&=3\\
2\cdot3+1&=7\\
2\cdot3\cdot5+1&=31\\
2\cdot3\cdot5\cdot7+1&=211\\
2\cdot3\cdot5\cdot7\cdot11+1&=2311\\
2\cdot3\cdot5\cdot7\cdot11\cdot13+1&=59\cdot509\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17+1&=19\cdot97\cdot277\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19+1&=347\cdot27953\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23+1&=317\cdot703763\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29+1&=331\cdot571\cdot34231\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31+1&=200560490131\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37+1&=181\cdot60611\cdot676421\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41+1&=61\cdot450451\cdot11072701\\
2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdot23\cdot29\cdot31\cdot37\cdot41\cdot43+1&=167\cdot78339888213593
\end{aligned}
$$
Is it true that all prime factors occur with multiplicity one in this list?
(Note that if one multiplies consecutive primes not starting from 2 and adds 1, there are many examples of multiplicities greater than one.)
Another question, probably much harder to answer: there are six primes in this list, the last one being $2\cdot3\cdot...\cdot31+1$. I've checked until $2\cdot3\cdot...\cdot227+1$ there are no primes, the number of prime factors slowly grows (first time that 5 factors occur is at $2\cdot3\cdot...\cdot127+1$, first time 6 factors occur is at $2\cdot3\cdot...\cdot137+1$, first time 7 factors occur at $2\cdot3\cdot...\cdot211+1$).
Are there any more primes in this list?

REPLY [7 votes]: There are more Euclid primes, but it isn't known if there are infinitely many. It's just conjectured, as well as all of Euclid numbers being squarefree: https://oeis.org/A006862