A question about numbers from Euclid's proof of infinitude of primes
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?
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
The product of the first $75$ primes, plus $1$, is prime. (That number is $171962010545840643348334056831754301958457563589574256043877$ $110505832165523856261308397965147955578800999455782202456522$ $6932906295208262756822275663694111$.)
(I misunderstood the question at this point. The poster wants to know if any prime appears more than once in any given entry of the sequence, not in any pair of entries in the sequence.)
$277$ is a factor of the seventh number ($510511$) and the seventeenth ($1922760350154212639071$).