I wanted to report on a variation, Gaussian prime random walks. As before start with any point in the complex plane $c_0$. Walk in the $+x$ direction until you hit a Gaussian prime. Unlike the original question, which requires a $90^\circ$ counterclockwise turn, turn left or right by $90^\circ$ with equal probability. Continue turning randomly $\pm 90^\circ$ at every Gaussian prime hit, until some Gaussian prime is hit for a second time. Call that a Gaussian prime loop.

Below is an example:

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          [![Loop62][1]][1]
      ^{Green: start at $c=100 + 2i$. Red: looped at $197 + 22 i$. Yellow: Gaussian prime turn.}

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Although this runs into the same open-problem impediment cited by @FrançoisBrunault,^* it appears that, for a random starting $c_0$ within some fixed distance of the origin, the probability of eventually looping might be $1$. Looping is much "easier" to achieve than is spiral cycling, which requires revisiting $c_0$ from the left.

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>^* _{"It's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$. So starting at $N+i$ and moving $+x$, we cannot exclude the possibility of hitting no prime."}

I wanted to report on a variation, Gaussian prime random walks. As before start with any point in the complex plane $c_0$. Walk in the $+x$ direction until you hit a Gaussian prime. Unlike the original question, which requires a $90^\circ$ counterclockwise turn, turn left or right by $90^\circ$ with equal probability. Continue turning randomly $\pm 90^\circ$ at every Gaussian prime hit, until some Gaussian prime is hit for a second time. Call that a Gaussian prime loop.

Below is an example:

---

          [![Loop62][1]][1]
      ^{Green: start at $c=100 + 2i$. Red: looped at $197 + 22 i$. Yellow: Gaussian prime turn.}

---

Although this runs into the same open-problem impediment cited by @FrançoisBrunault,^* it appears that, for a random starting $c_0$ within some fixed distance of the origin, the probability of eventually looping might be $1$. Looping is much "easier" to achieve than is spiral cycling, which requires revisiting $c_0$ from the left, whereas looping just needs to revisit some prime twice.

---

>^* _{"It's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$. So starting at $N+i$ and moving $+x$, we cannot exclude the possibility of hitting no prime."}

I wanted to report on a variation, Gaussian prime random walks. As before start with any point in the complex plane $c$. Walk in the $+x$ direction until you hit a Gaussian prime. Unlike the original question, which requires a $90^\circ$ counterclockwise turn, turn left or right by $90^\circ$ with equal probability. Continue turning randomly $\pm 90^\circ$ at every Gaussian prime hit, until some Gaussian prime is hit for a second time. Call that a Gaussian prime loop.

Below is an example:

---

          [![Loop62][1]][1]
      ^{Green: start at $c=100 + 2i$. Red: looped at $197 + 22 i$. Yellow: Gaussian prime turn.}

---

Although this runs into the same open-problem impediment cited by @FrançoisBrunault,^* it appears that, for a random starting $c$ within some fixed distance of the origin, the probability of eventually looping might be $1$. Looping is much "easier" to achieve than is spiral cycling, which requires revisiting a prime from the same direction.

---

>^* _{"It's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$. So starting at $N+i$ and moving $+x$, we cannot exclude the possibility of hitting no prime."}

I wanted to report on a variation, Gaussian prime random walks. As before start with any point in the complex plane $c_0$. Walk in the $+x$ direction until you hit a Gaussian prime. Unlike the original question, which requires a $90^\circ$ counterclockwise turn, turn left or right by $90^\circ$ with equal probability. Continue turning randomly $\pm 90^\circ$ at every Gaussian prime hit, until some Gaussian prime is hit for a second time. Call that a Gaussian prime loop.

Below is an example:

---

          [![Loop62][1]][1]
      ^{Green: start at $c=100 + 2i$. Red: looped at $197 + 22 i$. Yellow: Gaussian prime turn.}

---

Although this runs into the same open-problem impediment cited by @FrançoisBrunault,^* it appears that, for a random starting $c_0$ within some fixed distance of the origin, the probability of eventually looping might be $1$. Looping is much "easier" to achieve than is spiral cycling, which requires revisiting $c_0$ from the left.

---

>^* _{"It's unknown whether there are infinitely many Gaussian primes of the form $n+i$ with $n \in \mathbb{Z}$. So starting at $N+i$ and moving $+x$, we cannot exclude the possibility of hitting no prime."}