First-year graduateComputationPlanned
Experimental mathematics: conjectures and counterexamples
Using computation to discover patterns, generate conjectures and search for counterexamples: integer sequences and the OEIS, integer-relation algorithms such as PSLQ, exhaustive search, and computer-assisted proofs, with honest reporting of evidence.
math.comp.experimental-mathematics
The lesson for this topic has not been written yet. The outline below is part of the curriculum map and has not been reviewed.
Builds on
Required prerequisites, all the way down: 43
You will learn to
- Generate and test conjectures numerically, and design a counterexample search with clearly stated bounds.
- Use high-precision arithmetic and integer-relation detection to guess closed forms, and then look for proofs.
- Assess computer-assisted proofs, such as those of the four colour theorem and the Kepler conjecture, and how they were verified.
Definition
- An integer relation among real numbers x₁, …, xₙ is a non-zero integer vector (a₁, …, aₙ) with a₁x₁ + ⋯ + aₙxₙ = 0.
- A counterexample search is exhaustive up to a bound if it checks every case below that bound.
Theorem
- The Bailey–Borwein–Plouffe formula for π was found with the help of the PSLQ integer-relation algorithm.
- Hales's proof of the Kepler conjecture was formally verified by the Flyspeck project, completed in 2014.
- Numerical evidence can mislead: π(x) < li(x) for every x that has been checked, yet Littlewood proved that the inequality reverses infinitely often.
Common misconceptions
- Treating a pattern that holds in many cases as established.
- Reporting a search as exhaustive without stating its bounds.
Optional discoveries
- Research ethics and attribution Experimental results must be reported honestly, with code and bounds, and credited properly.
- ← Programming for mathematics
- ← Reading the mathematical literature
Learning pathways
Further reading
- Jonathan Borwein and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004)