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I'm looking to see if there is proof of the infinitude of prime numbers using a probabilistic method. I am motivated by the answer of my question here. The answer is based on a relationship between independence of measurable sets and coprime integers.

More precisely I am asking the following.

QUESTION: Is there a proof of the infinitude of primes using the Lovász local lemma by any of several different versions of the lemma?

John Baez
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Elias Costa
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    Why would one want to use LLL here? – Did Jul 03 '13 at 19:08
  • I put a "comment" after the answer by nullUser calling attention to this question. – Will Jagy Jul 03 '13 at 19:15
  • Note that, as you both are anonymous, one good choice would be a chat room for the two of you. That would work pretty well if you are in similar time zones. There are instructions on the longest running chat room about how to enable "ChatJax" in order to use Latex. – Will Jagy Jul 03 '13 at 19:18
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    @Did The beauty and richness of the proof of this theorem with the LLL would not be a sufficient reason? – Elias Costa Jul 03 '13 at 19:27
  • Which proof using the LLL? You might also ask for a proof of the CLT using the nine-point circle... – Did Jul 03 '13 at 19:36
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    This seems to be the stuff some rather nice, innovative papers are made of: somebody wants to prove something using methods and stuff that bear no straightforward, clear conection to that something...Of course, it could perfectly well be that nobody can (at least so far) see any possible conection between things and nothing happens. – DonAntonio Jul 03 '13 at 19:53
  • I know this question may seem nebulous. But I have strong motivations for this question. But if someone has already thought about this and had success on it this was almost certainly someone Paul Erdős. He may have had a positive success in the direction of the question but never published anything. But if there is something published about it I would know. – Elias Costa Jul 03 '13 at 20:09

2 Answers2

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I don't know how to do it using your proposed lemmas, but if you would like a probabilistic proof, we can work it out from my previous answer. Again take $P(X=n) = n^{-s}/\zeta(s)$ and $E_k := \{X \text{ is divisible by } k\}$. We already showed for $s>1$ $$ \left(\sum_{n=1}^\infty n^{-s}\right)^{-1}=\frac{1}{\zeta(s)} = P(X=1) = P(\cap_{p} E_p^c) = \prod_p(1-P(E_p)) = \prod_p(1-p^{-s}). $$

Assume for contradiction that there are finitely many primes. Now let $s\to 1^+$. Then we get $$ 0 = \prod_p(1-p^{-1}) $$ which cannot be, as the right side is a finite product of strictly positive terms. Thus it must be that there are infinitely many primes.

nullUser
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    Hi, I was confused about one thing. How do you know $n^{-s}/\zeta(s)$ is a real number in $[0,1]$? Or is it okay to have a complex-valued probability? – Francis Begbie Jan 11 '16 at 10:36
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    While it is true that $\zeta(s)$ is real for $s \in [0,1)$ (not at $s=1$), this is not necessary information to the proof. We consider only real $s>1$, and take the limit from the right, so as to only use $s>1$ throughout. – nullUser Jan 15 '16 at 19:03
  • same as this? https://math.stackexchange.com/a/427920 – BCLC Apr 17 '21 at 10:11
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If your question is whether there exists yet a proof that applies the Lovász/Erdős Local Lemma to prove the prime infinitude. The answer is, no. The Lemma provides a method that helps on existence proofs and gives not a motivation towards infinitude of primes.

If I misunderstood your question then I would need to know the exact motivation behind your inspiration.

al-Hwarizmi
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  • I believe that the proof would be the type to show that for any natural number $N$ the likelihood of a prime greater than $N$ would be positive. – Elias Costa Jul 03 '13 at 19:31
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    The infinitude of primes is an existence statement - it states that, given any finite set of primes, there exists another prime. – Thomas Andrews Jul 03 '13 at 19:33
  • @Elias That kind of argument won't work, because primes aren't random. Either there is a prime or there isn't a prime greater than $n$. Look at the proof to the previous question, and you'll see it defines a very specific random variable $X$. – Thomas Andrews Jul 03 '13 at 19:36
  • @Thomas Andrew His argument (on my coment) seems to make sense to me. But as the use of LLL is a matter of defining a suitable probability space the question is not refuted. – Elias Costa Jul 03 '13 at 19:42
  • @Thomas Andrews your existence statement can not hold in this case because one can clearly show that given any set of primes gives argument that there exists another prime. So the events in this argumentation are not indpendent. This is in harmony also with the fact that primes are not randomly distributed in white noise scheme. – al-Hwarizmi Jul 03 '13 at 19:46
  • @Elias exactly! the question is not refuted as such. You need however to provide a proper probability space and that is in terms of the primes not yet possible. Moreover I guess you would need the answer to your question (the proof of the infinitude) in order to make knowing the probability space. – al-Hwarizmi Jul 03 '13 at 19:50
  • @al-Hawarizmi I did not give the vote down. – Elias Costa Jul 03 '13 at 19:52
  • @Elias thanks. Hope the answer helps. – al-Hwarizmi Jul 03 '13 at 19:54