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One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $,modulo another prime number, $ p$, experimentally seems to be slightly morelikely to be a nonquadratic residue than a quadratic residue. We thought itwould be interesting to model this residue bias as a "random" walk usingLegendre symbol values as steps. Such a model would allow us to easilyvisualize the bias. In addition, we would be able to extend our model to othernumber fields.  In this report, we first outline underlying theory and some motivations forour research. In the second section, we present our findings in the rationalprime numbers. We found evidence that Chebyshev's bias, if modeled as aLegendre symbol $ \left(\frac{q}{p}\right)$ walk, may be somewhat reduced byonly allowing $ q$ to iterate over primes with nonquadratic residue (mod $ 4$).In the final section, we extend our Legendre symbol walks to the Gaussianprimes and present our main findings. Let $ \pi_1 = \alpha+\beta i$ and $ \pi_2= \beta+\alpha i$. We observed strong ($ \pm$) correlations between GaussianLegendre 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 explanationof why, for some norms, the plots for $ \pi_1$ and $ \pi_2$ have strongpositive correlation, while, for other norms, the plots have strong negativecorrelation. We hope to have written in a way that makes our observationsaccessible to readers without prior formal training in number theory.

Modeling Chebyshev's Bias in the Gaussian Primes as a Random Walk