# 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?

• @eyeballfrog Actually it just occurred to me that I can google for it, and I now think maybe I should not ask it, what do you think? – მამუკა ჯიბლაძე Mar 23 '17 at 8:43
• It's certainly known that there are more primes in this list. See OEIS A014545 for the indices. – nickgard Mar 23 '17 at 8:46
• Always a good plan. Also I should re-ask if by "multiplicity", you mean that it appears more than once in the list, or that it appears squared in the factorization. – eyeballfrog Mar 23 '17 at 8:46
• Somewhat related Wikipedia articles: Euclid number and Primorial prime. – Martin Sleziak Mar 23 '17 at 15:00
• The question whether any row in your table contains a repeated prime factor (part of what you ask), is a duplicate of: Are Euclid numbers squarefree? – Jeppe Stig Nielsen Mar 24 '17 at 9:36

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

• Thanks for the precise answer. I wonder how many Euclid primes are actually known - on OEIS they are A018239, and there only one more is listed, 171962010545840643348334056831754301958457563589574256043877‌​11050583216552385626‌​13083979651479555788‌​00999455782202456522‌​69329062952082627568‌​22275663694111 – მამუკა ჯიბლაძე Mar 23 '17 at 9:07
• Oh no wait, at A014545 there is much bigger list, indices of Euclid primes turn out to be at least these: 0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237 – მამუკა ჯიბლაძე Mar 23 '17 at 9:09
• Yeah, and the largest one known gives a (probably loose) bound on how many are known. – Vincenzo Oliva Mar 23 '17 at 9:12
• @მამუკაჯიბლაძე A top 20 of such primes, by Caldwell, extracted from his general Largest Known Primes database, can be found at http://primes.utm.edu/top20/page.php?id=5. – Jeppe Stig Nielsen Mar 24 '17 at 10:00
• @მამუკაჯიბლაძე Also PrimeGrid has a search for such primes. It is currently on their PRPNet server. – Jeppe Stig Nielsen Mar 24 '17 at 10:40

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 ($1922760350154212639071$).

• I was actually asking about multiplicities in each of them separately, but now I googled and it seems to be an open question, as well as whether there are infinitely many primes in the list. Maybe I still won't delete the question since you have answered it already... – მამუკა ჯიბლაძე Mar 23 '17 at 8:46
• Oh, I see what you mean. The question is a good one and shouldn't be deleted, I think, even though my answer is inadequate. – Patrick Stevens Mar 23 '17 at 8:47
• :D Should have read more carefully. I thought the +1-guys are called primorials...Whatever... – MooS Mar 23 '17 at 9:03