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Statement

Setting (pp. 283--284). Section 6 considers walks along the Gaussian integers relatively prime to a fixed integer $N$, viewed modulo $N$ in the square $[0,N-1]\times[0,N-1]$, and assumes $N$ even. The reflections $(a,b)\mapsto(-a,b)$, $(a,b)\mapsto(b,a)$ and $(a,b)\mapsto(N/2-b,N/2-a)$ preserve coprimality to $N$ and generate 16 reflections that cut the square into 16 triangles. The fundamental triangle $F(N)$ is drawn in Figure 6 (p. 284) as the triangle with vertices $(0,0)$, $(N/2,0)$ and $(N/4,N/4)$; the set printed for it on p. 284, $\{(a,b):a\ge b,\ a\le N/2,\ a+b\le N/2\}$, omits the side $b\ge0$. The propositions of Section 6 name no step size; the surrounding text applies them to walks of a fixed step $k$ (Conjecture 6.1, p. 284, and Section 7).

Proposition 6.1 (p. 284, quoted). "There is a walk to infinity along Gaussian integers relatively prime to $N$ if and only if there is a path inside the triangle $F(N)$ that touches all 3 edges of the triangle."

Source. Ilan Vardi, Prime percolation, Experimental Mathematics 7 (1998), no. 3, 275--289, doi:10.1080/10586458.1998.10504373: Section 6, pp. 283--284. The edition read is identified on the source card.

Read depth. Claims checked: the setting and the statement were read clause by clause on the printed pages. The paper gives no proof.

Proof pointer

None in the paper beyond the remark (p. 284) that the fundamental square tiles the plane under the translations by $(N,0)$ and $(0,N)$, after which Propositions 6.1--6.3 are called clear.

Dependencies

None.

Bears on

• #952: the finite check behind the paper's deduction of Theorem 7.1 (no unbounded walk of step $\sqrt2$, with $N=130$). All but finitely many Gaussian primes are coprime to a given $N$, so finding no crossing path for one $N$ and one step rules out a walk to infinity along the Gaussian primes with that step; the paper notes (p. 284) that the $N$ needed should grow doubly exponentially in the step, so the method is feasible only for very small steps.