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Feedback, Control, and Distribution of Prime Numbers

by Susan Marshall and Donald Smith

Year of Award: 2014

Award: Allendoerfer

Publication Information: Mathematics Magazine, vol. 86, 2013, pp. 189-203.

Summary (adapted from the Prizes and Awards booklet for MathFest 2013)

In this article, the authors describe an unusual application of a technique of mathematical modeling, feedback and control, to a classical mystery of number theory, the distribution of primes. In a famous result due to Gauss, the density of primes is (approximately) inversely proportional to the natural logarithm. The differential equation below reasonably models the density of primes. Here \(f(x)\) represents the density of primes:

\(f^{\prime}(x) = \frac{f(x) f(\sqrt{x})}{2x}\)

Although this is a known application in differential equation literature, it appears to be largely forgotten in number theory. In the process of deriving this model, the authors give the reader a lively introduction to the theory of feedback and control. The authors note that the distribution of prime numbers has an element of randomness, yet it also stays on track, much like a feedback and control system.

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About the Authors (From the MathFest 2014 Prizes and Awards Booklet)

Susan H. Marshall received a BS in Mathematics from Wake Forest University in 1993, with a minor in Psychology. After a brief stint as a data analyst for the Hubble Space Telescope at Goddard Space Flight Center in Maryland, she returned to school and received a PhD in Mathematics from the University of Arizona in 2001. While in graduate school, Susan studied Arithmetic Geometry. She was a postdoctoral fellow at the University of Texas at Austin from 2001 - 2004. She is currently an Associate Professor of Mathematics at Monmouth University, where she has just completed her 10th year. She lives on the Jersey Shore with her husband (and colleague) David, and their two children Gillian and Dylan.

Donald R. (Bob) Smith received an AB in Physics (magna cum laude) from Cornell in 1969, a MS in Operations Research from Columbia University in 1974 and a PhD in Operations Research from the University of California at Berkeley in 1975. He was an Assistant Professor of Operations Research at Columbia University from 1975-1979, before working at Bell Laboratories as a Member of Technical Staff and a Supervisor from 1980-2001. After leaving Bell Laboratories, he joined the faculty at Monmouth University where he is currently an Associate Professor in the Management and Decision Science Department. Most of his publications are in Operations Research journals in the areas of stochastic processes. Bob has always been fascinated by prime numbers because they are a deterministic system with elements of apparent randomness but hidden control. He and his wife Pat have 3 grown children and two grandchildren. He is an avid cyclist averaging over 11K miles per year.

Subject classification(s): Algebra and Number Theory | Number Theory | Primes
Publication Date: 
Monday, August 11, 2014