Experimenting with Apéry Limits and WZ pairs

This article, dedicated with admiration in memory of Jon and Peter Borwein,illustrates by example, the power of experimental mathematics, so dear to them both, by experimenting with so-called Apéry limits and WZ pairs. In particular we prove a weaker form of an intriguing conjecture of Marc Chamberland and Armin Straub (in an article dedicated to Jon Borwein), and generate lots of new Apéry limits. We also rediscovered an infinite family of cubic irrationalities, that suggested very good effective irrationalitymeasures (lower than Liouville's generic 3), and that we conjectured to go down to the optimal 2. As it turned out, as pointed out by Paul Voutier (see the postscript kindly written by him), our conjectures follow from deep results in number theory. Nevertheless we believe that further experiments with our Maple programs would lead to new and interesting results.

Jonathan Michael Borwein 1951 − 2016: Life and Legacy

Jonathan M. Borwein (1951−2016) was a prolific mathematician whose career spanned several countries(UK, Canada, USA, Australia) and whose many interests includedanalysis, optimization, number theory, special functions, experimental mathematics, mathematical finance, mathematical education,and visualization. We describe his life and legacy, and give anannotated bibliography of some of his most significant books and papers.

TEACHING PROOF TO STUDENTS USING EXPERIMENTAL MATHEMATICS

This article focuses on teaching students how to use experimental mathematics in proving mathematical proofs. Firstly, the proofs of the theorem are analyzed by experiment and as a result of the ability of intuitive thinking, it is proved analytically. In the process of training, using experiments will increase the quality of the educational system.

Automatic discovery of irrationality proofs and irrationality measures

We illustrate the power of Experimental Mathematics and Symbolic Computation to suggest irrationality proofs of natural constants, and the determination of their irrationality measures. Sometimes such proofs can be fully automated, but sometimes there is still need for a human touch.