  One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $,
modulo another prime number, $ p$, experimentally seems to be slightly more
likely to be a nonquadratic residue than a quadratic residue. We thought it
would be interesting to model this residue bias as a "random" walk using
Legendre symbol values as steps. Such a model would allow us to easily
visualize the bias. In addition, we would be able to extend our model to other
number fields.
  In this report, we first outline underlying theory and some motivations for
our research. In the second section, we present our findings in the rational
prime numbers. We found evidence that Chebyshev's bias, if modeled as a
Legendre symbol $ \left(\frac{q}{p}\right)$ walk, may be somewhat reduced by
only allowing $ q$ to iterate over primes with nonquadratic residue (mod $ 4$).
In the final section, we extend our Legendre symbol walks to the Gaussian
primes and present our main findings. Let $ \pi_1 = \alpha+\beta i$ and $ \pi_2
= \beta+\alpha i$. We observed strong ($ \pm$) correlations between Gaussian
Legendre symbol walks for $ \left[\frac{a+bi}{\pi_1}\right]$ and $
\left[\frac{a+bi}{\pi_2}\right]$ where $ N(\pi_1) = N(\pi_2)$ and $ a+bi$
iterates over Gaussian primes in the first quadrant. We attempt an explanation
of why, for some norms, the plots for $ \pi_1$ and $ \pi_2$ have strong
positive correlation, while, for other norms, the plots have strong negative
correlation. We hope to have written in a way that makes our observations
accessible to readers without prior formal training in number theory.
