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Let $n$ be an integer parameter. Is there a polynomial $f(x,y,n)\in \mathbb{Z}[x,y,n]$ such that $$ n\in \mathbb{N} \Longleftrightarrow \exists x,y\in \mathbb{Z}, f(x,y,n)=0? $$

This is a generalized form of the infamous Poonen Christmas problem. Bjorn's question boils down to the special case when $f(x,y,n)$ is of the form $cn+g(x,y)$ for some nonzero integer $c$ and some $g(x,y)\in \mathbb{Z}[x,y]$.

A formula by Raphael Robinson in Arithmetical definitions in the ring of integers, modified slightly as by Zhi-Wei Sun in A new relation-combining theorem and its applications (and reiterated by Mihai Prunescu in this paper) gives the following near example: $$ n\in \mathbb{N} \Longleftrightarrow \exists x,y\in \mathbb{Z}, (x^2-(4n+2)y^2-1=0) \land (y\neq 0). $$ "Open Question A" in Prunescu's linked paper is exactly my question.

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    $\begingroup$ The trick $n\in \mathbb N\iff \exists x,y\in\mathbb Z(x^2-(4n+2)y^2=1\land y\not=0)$ first appeared in my 1992 paper available from maths.nju.edu.cn/~zwsun/14z.pdf $\endgroup$ Commented Oct 13 at 14:57
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    $\begingroup$ The question you asked is a known open problem about diophantine representation of $\mathbb N$ over $\mathbb Z$ in fewer unknowns. Such things are closely related to Hilbert's Tenth Problem over $\mathbb Z$ in few unknowns. See link.springer.com/article/10.1007/s11425-020-1813-5 . $\endgroup$ Commented Oct 13 at 15:02
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    $\begingroup$ As someone who has thought about Poonen's question seriously, this is the first time I've heard it being referred to as Poonen's Christmas question. I will be sure to use that phrase in future works on the problem! $\endgroup$ Commented Oct 13 at 16:35
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    $\begingroup$ @TimothyChow absolutely, and this is acknowledged in our paper (ems.press/journals/jems/articles/14299022). However, for us it was Poonen's MO question that compelled us to work on it (Poonen, in private communication, told us he was not aware of Lew asking this question earlier). $\endgroup$ Commented Oct 14 at 1:24
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    $\begingroup$ I think you could apply some of the methods I describe in my (conditional) solution to Poonen's Christmas Problem (see my answer there). The first half should go through, looking at the surfaces f(x, y, n^m) = 0, as m increases, so integral points should be dense. From the second half, I think you also get that (the projective closure of) f(x, y, n) = 0 is a projective line over Q(n). Perhaps these constraints are helpful for anyone searching for an example. $\endgroup$ Commented Oct 15 at 9:26

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