Experimental Mathematics and Integer Relations
by Jonathan M. Borwein
The emergence of powerful mathematical computing environments, growing availability of correspondingly powerful (multi-processor) computers and the pervasive presence of the Internet allow mathematicians to proceed heuristically and quasi-inductively. Its exciting consequences have been studied over the past ten years at the Centre for Experimental and Constructive Mathematics of Simon Fraser University, Canada. They include several recent results connected with the Riemann Zeta Function.
Mathematicians increasingly use symbolic and numeric computation, visualisation tools, simulation and data mining. This is both problematic and challenging. For example, we mathematicians care more about the reliability of our literature than other sciences. These new developments, however, have led to the role of proof in mathematics now being under siege.
Many of my favourite examples originate in between mathematical physics and number theory/analysis/knot theory and involve the ubiquitous Zeta Function, of Riemann hypothesis fame. They rely on the use of Integer Relations Algorithms: A vector (x1, x2,..., xn) of real numbers possesses an integer relation if there are integers ai not all zero with
0 = a1 x1+ a2 x2 + ... + an xn.
The goal is to find ai if such exist, and if not found, to obtain lower bounds on the size of possible ai.
For n = 2, Euclids algorithm gives the solution. The first general algorithm was found in 1977 by Ferguson and Forcade, followed by Lenstra, Lenstra and Lovász (LLL) in 1982, implemented in Maple and Mathematica, and my favourite PSLQ (Partial Sums using matrix LQ decomposition) in 1991, with a parallel version in 1999. Recently, J. Dongarra and F. Sullivan ranked Integer Relation Detection among the 10 algorithms with the greatest influence on the development and practice of science and engineering in the 20th century. (Some others: Monte Carlo, Simplex, FFT.)
A CECM interface allows one to find relations and explore the underlying algorithms. Three examples are presented in the Boxes: 1. Algebraicness, 2. Riemann Zeta Function, 3. Binomial Sums.
Results in Box 3 were obtained via Gospers (Wilf-Zeilberger type) telescoping algorithm (see also Paules contribution in this issue). Identity (3) was recently proved by Almkvist and Granville (Experimental Math, 1999) thus finishing the proof of (2) and perhaps shedding light on the irrationality of (7)? (2N+1) is not proven irrational for N > 1, though it is now known that one of (3,5,7,9,11) is!
Many other similar triumphs, and an equal number of failures are described in the references given below, in fields as diverse as quantum field theory, nuclear magnetic resonancing, probability, complexity, algorithm design, and coding theory: