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EDIT: I forgot how open number theory is! (I think that gets me put on mathematician probation or something.) For this question, I will accept any answer which assumes "standard conjectures" such as RH, Schanuel, Bunyakovsky, etc. I'm definitely more optimistic about conditional results.

Say that a factonomial sequence is a (possibly infinite) sequence of natural numbers $x_i$ such that

  • each $x_i$ is prime, and

  • there is some (single variable, integer coefficients, nonconstant) polynomial $p$ such that for all $i>1$ we have $$x_i=p(\prod_{j<i}x_j).$$

Call the polynomial $p$ the shape of the factonomial sequence; each factonomial sequence is determined by its shape and its initial value.

The basic idea is that factonomial sequences come out of some elementary proofs that infinitely many primes of a certain form exist. For instance, the usual "multiply everything and add one" argument gives rise to the shape $p_1(u)=u+1$, and the usual "multiply everything twice and add two" proof that there are infinitely many primes $\equiv 3 (\mathsf{mod}$ $4)$ gives rise to the shape $p_2(u)=u^2+2$. The maximal factonomial sequence with shape $p_2$ and starting value $3$ is $$3,11,1091, 1296216011, 2177870960662059587828905091.$$ The next term would be $$10329907495268194677701503661780370732730049826819138974714891651071966324541232011,$$ but that's not prime. (Amusingly, its smallest prime factor is $41$, but its smallest prime factor which is $3$ mod $4$ is a bit bigger: $76870667$.)

Question: is there an infinite factonomial sequence?

(This is a question which I'm 99% sure one of my students will ask me in the next couple days!) I'm aware of multiple results of the form "no function of such-and-such type has output consisting entirely of primes," but I don't immediately see one which applies here. Unfortunately, terms in factonomial sequences grow so fast that I can't do much experimenting.

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    $\begingroup$ The only sequences growing so quickly and known to produce infinite many primes, should be of this kind. We can surely construct numbers this way without small prime factors, but if a number is large , the chance is still very small that it is prime. So, my guess is that there is no such sequence, but such statements usually are completely out of reach to be proven or disproven. $\endgroup$
    – Peter
    May 18 at 9:16
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    $\begingroup$ @Peter "The only sequences growing so quickly and known to produce infinite many primes, should be of this kind." Why do you say that? That seems like a strong claim. $\endgroup$
    – Noah Schweber
    May 19 at 3:49
  • $\begingroup$ U don't think that it is known that there exists a polynomial $p:\Bbb N\to \Bbb Z$ with integer co-efficients, such that (i)deg $p>1$ and (ii) $p(n)$ is prime for infinitely many $n\in\Bbb N$....mathworld.wolfram.com/BouniakowskyConjecture.html and mathworld.wolfram.com/Prime-GeneratingPolynomial.html $\endgroup$
    – DanielWainfleet
    May 21 at 7:02
  • $\begingroup$ I believe this is another one of those NT questions that are extremely hard to prove or disprove... $\endgroup$
    – Shengtong Zhang
    May 22 at 17:32
  • $\begingroup$ @DanielWainfleet That's a fair point - I'd forgotten that even that statement was open! I've edited the question to just ask for conditional results. $\endgroup$
    – Noah Schweber
    May 22 at 19:03

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