Experimental mathematics: conjectures and counterexamples

Se basa en

Requisitos previos obligatorios, hasta el fondo: 43

Aprenderás a

• 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.

Definición

• 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.

Teorema

• 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.

Errores frecuentes

• Treating a pattern that holds in many cases as established.
• Reporting a search as exhaustive without stating its bounds.

Descubrimientos opcionales

• Research ethics and attribution Experimental results must be reported honestly, with code and bounds, and credited properly.
• ← Programming for mathematics
• ← Reading the mathematical literature

Itinerarios de aprendizaje

• Research preparation

Lecturas complementarias

• Jonathan Borwein and David Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century (2004)