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Tapas in Experimental Mathematics

Tewodros Amdeberhan and Victor H. Moll, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
Contemporary Mathematics 457
[Reviewed by
William J. Satzer
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Tapas, as the reader may know, are small snacks often served as an appetizer and sometimes accompanied by alcoholic beverages. The tapas in this volume provide a more intellectual kind of nourishment (with or without alcohol). They include eighteen papers on the subject of experimental mathematics, which — according to the preface — involves the use of advanced computing techniques in mathematical research. To be more specific, computation in experimental mathematics is used for:

  • Gaining insight and intuition.
  • Discovering patterns and relationships.
  • Using graphical displays to suggest underlying mathematical principles.
  • Testing (and especially falsifying) conjectures.
  • Exploring a possible result to see if it merits a formal proof.
  • Suggesting approaches for a formal proof.
  • Replacing lengthy by-hand derivations with computer-based ones.
  • Confirming analytically derived results.

I have included the editors’ complete list because the term “experimental mathematics” is new and its scope is not yet so clear. The papers in this volume cover a wide range and encompass all the items above and more. “Advanced computing techniques” generally means the use of a digital computer and associated software. Of course, one could argue that Gauss, for example, did experimental mathematics when he studied the distribution of primes; he scrutinized Schulze’s table of logarithms, extended the tables, tabulated primes in various ranges, and conjectured what became the Prime Number Theorem.  Nonetheless, it’s clear that modern computing power together with high quality commercial software (such as Mathematica, Maple and MATLAB) offers dramatically enhanced capabilities. In addition, there is a growing collection of specialized software such as Helaman Ferguson’s PSLQ. The PLSQ software is designed to detect relationships among integers: For a set of real numbers {x1, ..., xn}, the algorithm looks for integers {a1, ..., an} such that a1x1 + ... + anxn = 0. A high degree of numerical precision is necessary to avoid spurious relationships signaled by large values of the integers ai.

The volume under review is part of the AMS Contemporary Mathematics series and includes proceedings of the AMS Special Session on Experimental Mathematics from January 2007.  A paper by David Bailey and Jonathan Borwein (two of the earliest advocates and practitioners of this form of experimental mathematics) is the closest thing in this volume to a survey; it’s called “Computer-Assisted Discovery and Proof”. They discuss the principal algorithms used in experimental and computer-assisted mathematics. These include high-precision integer and floating point arithmetic, high precision evaluation of integrals and infinite series and symbolic computation. They also briefly discuss the Wilf-Zeilberger algorithm, a non-numerical algorithm that uses “creative telescoping” to show that a sum (with finitely or infinitely many terms) is zero.

Most of the other papers in this collection focus on relatively narrow areas while demonstrating variations of the previously described techniques. A couple of items of note include a paper by Berndt et al. discussing questionable claims found in Ramanujan’s lost notebook and their attempt to prove, correct, or give proper interpretations to those claims. A paper by Ayyer and Zeilberger examines a combinatorial problem involving two-dimensional lattice walks with boundaries. The work here was motivated by the statistical mechanical study of polymers held between two close plates.

Although it is rather specialized, this book is worth dipping into for any reader interested in the subject of experimental mathematics. A related textbook, also worth a look, is Mathematics by Experiment, Plausible Reasoning in the 21st Century, by Borwein and Bailey.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.