"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.

lotof 'primes' without the integrality condition. But this is just a crazy stab, and probably nothing. $\endgroup$ – David Roberts Sep 5 '11 at 2:18