
A Curious Sequence of Prime Numbers - headalgorithm
https://blogs.scientificamerican.com/roots-of-unity/a-curious-sequence-of-prime-numbers/
======
_Microft
Isn't _" A common misconception about this proof is that the number
p_1×p_2×p_3×…×p_n +1 itself has to be prime."_ plain wrong? They seem to
assume that this product generates p_n+1 while it would generate p_m with m >=
n + 1 instead, thereby maybe leaving gaps in the sequence (making attempts to
use the method again and again to generate all primes unsuccessful).

Edit: the counter-example of 2 x 3 x 5 x 7 x 11 x 13 + 1 = 59 * 509 proves
this wrong. My mistake.

~~~
glial
I thought it was wrong -- but looking back at Euclid's proof, Euclid doesn't
specify that p_1...p_n be all the primes -- just all the _known_ primes, so
there could be gaps. He says that either

(1) -- p_1×p_2×p_3×…×p_n +1 is prime, or

(2) -- p_1×p_2×p_3×…×p_n +1 is not prime, but it's also not divisible by any
of the pre-specified primes, so there must be another prime that divides into
it aside from p_1...p_n.

See here:
[https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/propI...](https://mathcs.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html)

~~~
_Microft
OK, I understand. Strange that the non-successive primes are numbered with
successive indices though.

PS: Well, actually not because they are not numbering all primes but numbering
primes they know.

~~~
kosievdmerwe
Even if they are successive then the product plus one might not be prime,
since the product plus one is so much larger than p_n, there might be a prime
that divides it (and likely at least 2).

The first non-prime example is 2x3x5x7x11x13 + 1 which equals 59 x 509

~~~
_Microft
Sure, see my first post about that. I amended it shortly after I posted but it
felt wrong to remove the part that people responded too already.

------
pahkah
At first I read

> The similar sequence that chooses the largest instead of smallest prime that
> divides 1 plus the product of the previous terms avoids an infinite number
> of primes.

as saying that it avoids having an infinite number of primes in the sequence
(i.e. that the sequence itself contained a finite number of distinct primes),
but it's actually that there's an infinite number of primes not present in the
sequence.

Fascinating sequence, good article, thanks!

~~~
sp332
Oh thanks, I read it that way too.

------
JadeNB
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.

~~~
human20190310
I only just now understood that Euclid's method doesn't necessarily generate a
prime number; it produces a number that you can't "get to" by multiplying the
primes in your list of prime numbers; as such any list of primes is
incomplete.

~~~
eridius
And since you can't "get to" it with your existing list, this means the prime
factorization of this new number includes at least one prime not on your list,
and therefore you can grow your list with every step (assuming you're capable
of prime factorizing arbitrary numbers).

