Gauss Sum Walks

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.
  • Contributed by: Erqian Wang
  • (Based on an undergraduate research project at the Illinois Geometry Lab by Ka Fung Tjin, Erqian Wang and Yifan Zhang and directed by A. J. Hildebrand.)

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DETAILS

Given an integer , the exponential sum version of the Gauss sum modulo is defined by
and the quadratic residue version is defined by
,
where is the Jacobi symbol, which has values 0, 1, depending on the quadratic residue properties of modulo .
A famous result of Gauss states that when the modulus is a prime , the two sums and have the same value , given by
.
For composite moduli , the two sums are in general different, and the above formula does not necessarily hold.
It was shown by Lehmer [1] that the exponential Gauss walk always has the spiral-type shape observed in the visualization. On the other hand, the shape of the quadratic residue Gauss walk is more mysterious and has not been explored in the literature.
Reference
[1] D. H. Lehmer, "Incomplete Gauss Sums," Mathematika, 23(2), 1976 pp. 125–135. doi:10.1112/S0025579300008718.
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