"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.
Best Answer
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.