A way less interesting comment on the Euclid–Mullin sequence, for anyone who's just learned about it in a math class: it is common to take away the wrong lesso

A way less interesting comment on the Euclid–Mullin sequence, for anyone who's just learned about it in a math class: it is common to take away the wrong lesson "the product of all the primes so far, plus 1, is always prime." (Even if it's totally obvious to you that this is false, I assure you that it's not obvious to everyone. Sigh, I've also just noticed that, by skipping to the part of the article about interesting unsolved questions, I missed that the author says exactly this. Sorry! Hopefully the rest of the comment is worth something.) This succeeds for a little while, since (the product of no primes) + 1 is 2, 2 + 1 is 3, 2·3 + 1 is 7, and 2·3·7 + 1 is 43, and most math geeks will recognise these as primes. However, it fails immediately after: 2·3·7·43 + 1 is 13·139. (If you take "all the primes so far" to mean not just "all the primes generated this way" but "all primes less than or equal to those generated this way", then we get a slightly different trajectory: after 7 comes 2·3·5·7 + 1 = 211, which is prime; and then the so-called primorial (https://en.wikipedia.org/wiki/Primorial https://en.wikipedia.org/wiki/Primorial) 211# has a successor that is huge, way bigger than 2297, but is divisible by 2297.) It is possible to modify the sequence so that it proveably generates all primes; see, for example, Booker - A variant of the Euclid–Mullin sequence containing every prime (https://arxiv.org/abs/1605.08929 https://arxiv.org/abs/1605.08929). According to Wikipedia, the notion was introduced in Mullin - Recursive function theory (A modern look at a Euclidean idea) (https://doi.org/10.1090%2FS0002-9904-1963-11017-4 https://doi.org/10.1090%2FS0002-9904-1963-11017-4). EDIT: Replying to child (https://news.ycombinator.com/item?id=20017324 https://news.ycombinator.com/item?id=20017324 )—sorry for doing it here; I'm "posting too fast", but don't want to ignore it: > I'm genuinely confused by this. I don't see how 2·3·7·43 is "the product of all the primes so far", since 11, 13, 17, 19, 23, 29, 31, 37, and 41 are missing. > Does the author mean "all the primes generated by the sequence", or "all the primes up to a certain number"? I always thought the Euclid method involved the latter. Just to be clear, I'm not the author of the linked post. I did write "all the primes so far" to mean "all the primes generated in this way", which is the sense in which Mullin used it in the article linked above. Both approaches work to generate 'new' primes, in the sense that they're not among the primes multiplied together; but you must use the primorial if you want to generate new, bigger primes. I edited to take into account the more natural interpretation (see the parenthetical above), but probably it was while you were posting.