The alternate case in Euclid's proof about infinitude of primes

Question

In Euclid's famous proof that there is no largest prime number he supposes $p_n$ is the largest prime and forms:

$$\prod_{i=1}^n p_i + 1$$

and notes that the product must be larger than any of its components and either a) this is a prime or b) it is the product of primes larger than $p_n$.

What is the first number where case b) holds? Are there an infinite number of these?

The first example is given at https://math.stackexchange.com/questions/1380543/why-is-it-that-the-product-of-first-n-prime-numbers-1-another-prime and I expect that the answer to b) is not known but is generally believed to be, yes. . — Gerry Myerson, May 04 '25 at 11:30
This question is similar to: Is there an infinite number of primes constructed as in Euclid's proof?. If you believe it’s different, please [edit] the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. — Robin, May 04 '25 at 15:26

Steven · Accepted Answer · 2025-05-04T13:25:55.237

The numbers $$E_k = \prod_{i=1}^k p_i + 1$$ are called Euclid numbers. The first indices $k$ such that $E_k$ is prime are $$1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865,$$ see OEIS A014545.

It is not known whether infinitely many Euclid numbers are prime, see the wiki for example. If you look at the indices above, it does seem like infinitely many Euclid numbers should be composite, but this also seems to be an open problem.

The alternate case in Euclid's proof about infinitude of primes

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