Pseudorandom Walks with Generalized Gauss Sums

exponent

2

3

4

5

6

7

modulus

4920

animate

axes

random modulus

This Demonstration shows pseudorandom walks constructed from generalized Gauss sums, defined by

m

∑

n=1

2πi

k

n

m

e

, where the modulus

m

and the exponent

k

are integers of at least 2. (The case

k=2

reduces to the classical quadratic Gauss sum.)

The random walks start at the origin, then the end point, as indicated by the yellow dot, takes steps given by the terms of the sum. The walks exhibit complicated behavior with curlicue patterns and sometimes unexpected symmetries.