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Gems of Experimental Mathematics

Tewodros Amdeberhan, Luis A. Medina, and Victor H. Moll, editors
American Mathematical Society
Publication Date: 
Number of Pages: 
Contemporary Mathematics 517
[Reviewed by
Charles Ashbacher
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Since the first abstract mathematical thought arose in a human brain, experimental mathematics has been a fundamental component of the advance in mathematical knowledge. Computations have been the leading indicator of most mathematical proofs; someone does a few, sees that a pattern emerges and then is able to generalize the pattern into a formal proof.

The difference in the modern world is the incredible power of symbolic mathematical software packages such as Maple and Mathematica. Using a few lines of code, a researcher can potentially run millions of tests in order to verify a hypothesis or identify and prove a perceived pattern.

This book contains articles about some very advanced examples of experimental mathematics in action. The topics cover a wide spectrum of topics, from finding differential equations of the Calabi-Yau form to generating functions of several forms to algorithms for finding the digits of π. Many of the articles contain statements of open conjectures, which is always the goal of mathematics. Namely, prove something and pose a further question. Experimental mathematics can give a strong hint, but it is the formal reasoning of a proof that completes the task.

Outside of the common thread of being examined using experimental tactics, there is no common thread to the topics in this book. Therefore, few people will be interested in all the articles, but most will be interested in a few.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

  • G. Almkvist -- The art of finding Calabi-Yau differential equations
  • T. Amdeberhan -- A note on a question due to A. Garsia
  • D. H. Bailey and J. M. Borwein -- Experimental computation with oscillatory integrals
  • D. H. Bailey, J. M. Borwein, D. Broadhurst, and W. Zudilin -- Experimental mathematics and mathematical physics
  • S. T. Boettner -- An extension of the parallel Risch algorithm
  • R. P. Boyer and W. M. Y. Goh -- Appell polynomials and their zero attractors
  • O-Y. Chan and D. Manna -- Congruences for Stirling numbers of the second kind
  • M. W. Coffey -- Expressions for harmonic number exponential generating functions
  • R. E. Crandall -- Theory of log-rational integrals
  • S. Garoufalidis and X. Sun -- A new algorithm for the recursion of hypergeometric multisums with improved universal denominator
  • I. Gonzalez, V. H. Moll, and A. Straub -- The method of brackets. Part 2: Examples and applications
  • J. G. Goyanesa -- History of the formulas and algorithms for $pi$
  • J. Guillera -- A matrix form of Ramanujan-type series for $1/pi$
  • K. Kohl and F. Stan -- An algorithmic approach to the Mellin transform method
  • C. Koutschan -- Eliminating human insight: An algorithmic proof of Stembridge's TSPP theorem
  • M. L. Lapidus and R. G. Niemeyer -- Towards the Koch snowflake fractal billiard: Computer experiments and mathematical conjectures
  • L. A. Medina and D. Zeilberger -- An experimental mathematics perspective on the old, and still open, question of when to stop?
  • M. J. Mossinghoff -- The distance to an irreducible polynomial
  • S. Northshield -- Square roots of 2 x 2 matrices
  • O. Oloa -- On a series of Ramanujan
  • P. Raff and D. Zeilberger -- Finite analogs of Szemerédi's theorems
  • A. V. Sills -- Towards an automation of the circle method
  • J. H. Silverman -- The greatest common divisor of $a^n-1$ and $b^n-1$ and the Ailon-Rudnick conjecture
  • J. Sondow and K. Schalm -- Which partial sums of the Taylor series for $e$ are convergents to $e$? (and a link to the primes 2, 5, 13, 37, 463). II
  • C. Hillar, L. García-Puente, A. M. Del Campo, J. Ruffo, Z. Teitler, S. L. Johnson, and F. Sottile -- Experimentation at the frontiers of reality in Schubert calculus
  • Y. Yang and W. Zudilin -- On Sp$_4$ modularity of Picard-Fuchs differential equations for Calabi-Yau threefolds