"One cannot walk to infinity on the real line if one uses steps of bounded length and steps on the prime numbers. This is simply a restatement of the classic result that there are arbitrarily large gaps in the primes."

So begins the paper by Gethner, Wagon, and Wick, "A Stroll Through the Gaussian Primes" (American Mathematical Monthly 105(4): 327-337 (1998).) They explain that it is unknown if one can walk to infinity on the Gaussian primes with steps of bounded length. Paul Erdős was reported to have conjectured this is possible ("A conjecture of Paul Erdős concerning Gaussian primes." Math. Comp 24: 221-223 (1970); PDF). Later Erdős is reported to have conjectured the opposite: that no such walk-to-$\infty$ is possible [GWW98, p.327]. This has become known as the Gaussian Moat Problem, apparently still unresolved.

My question is:

Is there an analogous Quaternion Moat Problem? Is it solved? Open? Is it easier or harder than the Gaussian Moat Problem?

Define a nonzero quaternion $q = a + bi + cj + dk$ as prime prime iff (a) it is a Hurwitz quaternion (all components integer, or all components half-integer) and (b) its norm $a^2 + b^2 +c^2 + d^2$ is prime. (Part (b) is a consequence of the inability to factor $q$; see, e.g., Theorem 15 in "A Proof of Lagrange's Four Square Theorem Using Quaternion Algebras." Drew Stokesbary, 2007; PDF).

Can one "walk-to-$\infty$" on the quaternion primes using steps of bounded length?

Perhaps relevant here is Langrange's four-square theorem, which states that any natural number can be represented as the sum of four squares.

I ask this question in relative naïveté, and appreciate being enlightened.

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    $\begingroup$ You're using the word "quaternion" loosely, as if "real number" meant "integer". You intended to impose some kind of integrality constraint on the coefficients of your quaternions. Do you mean to focus on the quaternions with integral coefficients or the larger ring that allows all coefficients to be halves of odd integers too? Your discussion of the meaning of prime is also vague, as you don't make clear whether the properties you describe are definitions or genuine results (and for which kind of integral quaternion)? $\endgroup$ – KConrad Sep 5 '11 at 0:34
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    $\begingroup$ I don't think you can avoid the Hurwitz restriction, e.g., consider 4+(2i+2j+k)/3 which has norm 17. $\endgroup$ – François G. Dorais Sep 5 '11 at 1:50
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    $\begingroup$ I suspect (but I haven't checked) that what you obtain with the prime norm criterion is probably dense everywhere and the problem becomes trivial. $\endgroup$ – François G. Dorais Sep 5 '11 at 2:12
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    $\begingroup$ a crazy idea: the unit quaternion sphere has some interesting folations that the unit complex numbers don't have. Unless you force some sort of integrality, given a 'prime' with integrality condition, you probably get a lot of 'primes' without the integrality condition. But this is just a crazy stab, and probably nothing. $\endgroup$ – David Roberts Sep 5 '11 at 2:18
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    $\begingroup$ One could ditch the quaternions and just ask whether it's possible to walk to infinity on points of prime squared-length in ${\bf Z}^4$, no? I mean, it's not the same question, but it might be just as interesting. $\endgroup$ – Gerry Myerson Sep 5 '11 at 5:43

Having an infinite walk of bounded step length in the quaternions (or in $\mathbb Z^k$ in Gerry's version), gives us a sequence of primes $p_1,p_2\dots$ with $p_{k+1}-p_k=O(\sqrt{p_k})$. However the best unconditional result we have so far on prime gaps is $O(p_k^{0.525})$ by Baker, Harman and Pintz. So these problems are all open in general.

That said the heuristics that work for Gaussian primes can almost always be translated to a more general setting. One famous article on the topic is Vardi's paper "Prime percolation". There it is mentioned that the percolation model can be extended to the general case of primes represented by quadratic forms, quaternion primes etc. (where one can make the same predictions), though this is not written anywhere.

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    $\begingroup$ Here is the [BHP] paper to which you refer: "The Difference Between Consecutive Primes, II." Proceedings of the London Mathematical Society (2001), 83: 532-562. journals.cambridge.org/action/… $\endgroup$ – Joseph O'Rourke Sep 5 '11 at 13:46
  • $\begingroup$ What was not at first clear to me was your key observation that $p_{k+1}-p_k = O( p_k^{0.5} )$. It is a nice insight---very clever! Thanks! $\endgroup$ – Joseph O'Rourke Sep 6 '11 at 15:23

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