This Demonstration shows two types of pseudorandom walks constructed from Gauss sums, an exponential (red) and a quadratic residue (blue) Gauss walk. The random walk starts at the origin, takes steps given by the terms of the exponential or quadratic residue Gauss sum modulo , and ends at the value of the Gauss sum. By a famous result of Gauss, if the modulus is a prime number, the two walks always end at the same point, located at or depending on the remainder of modulo 4. The exponential Gauss walk has a characteristic shape consisting of two spirals. The quadratic residue Gauss walk exhibits a more complex behavior whose shape is roughly determined by the remainder of modulo 24.
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