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?
Best Answer
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