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Now, there are a couple of questions exactly about this topic already, but none of the answers I've read was satisfying to me. They are all about the fact that Willans' formula is numerically inefficient. Ok, I understand that, but for pure mathematicians it's not relevant whether a formula or an algorithm is numerically efficient or not, it just matters whether it's correct, and Willans' formula is. As far as I'm concerned, it is an explicit formula giving the $n$th prime in terms of elementary functions (plus the factorial); how isn't that relevant in number theory?

A rural reader
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Elvis
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    It's Willans. Not Willians. – lulu Jul 05 '24 at 00:32
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    You are right, the formula is correct. For mathematicians is not relevant not because is inefficient, but because is not useful to prove any properties about prime numbers. – jjagmath Jul 05 '24 at 00:35
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    See Conway's PrimeGame for another way to generate the list of primes. Again, very interesting, but not terribly useful either for generating the list of primes nor for proving theorems about primes. – lulu Jul 05 '24 at 00:44
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    It's useless because you can't use it for anything. Have you tried? Can you even prove that $p_n \to \infty$ from it? – Qiaochu Yuan Jul 05 '24 at 00:51
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    "it is an explicit formula giving the nth prime in terms of elementary functions (plus the factorial), how isn't that relevant in number theory" How is it relevant in number theory? Until you can use it to figure something out, it isn't relevant. – Noah Schweber Jul 05 '24 at 01:13
  • https://math.stackexchange.com/questions/4670123/why-is-willans-formula-for-primes-not-considered-a-valid-formula – mr_e_man Jul 05 '24 at 01:27
  • Complaints about Willan's formula go back to essentially it's introduction in the literature...as do defenses of it. Compare https://www.jstor.org/stable/3612863 and https://www.jstor.org/stable/3613607. – Semiclassical Jul 05 '24 at 01:28
  • The formula may be surprising at first glance but it is a special case of the fact that we can implement integer arithmetic (and Turing machines) using certain elementary functions. This is not useful for much of anything except for proving theoretical results such as certain problems on elementary function are undecidable (by using said implementation to encode unsolvable Diophantine problems e.g. see Richardson's paper cited here). Once you understand how such encodings work it all becomes utterly trivial. – Bill Dubuque Jul 05 '24 at 01:41
  • @lulu That "game" is just a standard prime enumeration algorithm implemented in Conway's Turing complete fraction-based programming language FRACTRAN. Beware that I discovered a bug in his original FRACTRAN program for computing $\pi,,$ see here. $\ \ $ – Bill Dubuque Jul 05 '24 at 02:08
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    @BillDubuque Thanks! Aware of FRACTRAN, and the $\pi$ computation...didn't know about the bug. Will read. – lulu Jul 05 '24 at 02:11
  • @Semiclassical It would be helpful to copy your comment to the duplicate since this might get deleted. – Bill Dubuque Jul 05 '24 at 06:59
  • @BillDubuque Good idea---done – Semiclassical Jul 05 '24 at 20:10
  • It's only useless, because no one has figured out yet how to "crack the formula" mathematically. – Daniel Donnelly Oct 28 '24 at 09:03

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