Experimental mathematics is a type of mathematical investigation in which computation is used to investigate mathematical structures and identify their fundamental properties and patterns. As in experimental science, experimental mathematics can be used to make mathematical predictions which can then be verified or falsified on the bases of additional computational experiments.
Borwein and Bailey (2003, pp. 2-3) use the term "experimental mathematics" to mean the methodology of doing mathematics that includes the use of computation for:
1. Gaining insight and intuition.
2. Discovering new patterns and relationships.
3. Using graphical displays to suggest underlying mathematical principles.
4. Testing and especially falsifying conjectures.
5. Exploring a possible result to see if it is worth formal proof.
6. Suggesting approaches for a formal proof.
7. Replacing lengthy hand derivations with computer-based derivations.
8. Confirming analytically derived results.
Examples of tools of experimental mathematics include computer algebra, symbolic algebra, Gröbner basis, integer relation algorithms (such as the LLL algorithm and PSLQ algorithm), arbitrary precision numerical evaluations, computer visualization, cellular automata and related structures, and databases of mathematical structures such as the Online Encyclopedia of Integer Sequences (https://oeis.org/) by Neil Sloane, The Wolfram Functions Site (https://functions.wolfram.com/) by Michael Trott and Oleg Marichev, and MathWorld (https://mathworld.wolfram.com/) by Eric Weisstein.
Artificial intelligence systems can participate in the same experimental cycle by proposing conjectures, searching examples and counterexamples, suggesting proof strategies, or helping to formalize arguments. Their output does not by itself replace mathematical proof. Generated claims must still be checked against definitions, computation, and formal or human verification. Since these activities are methods and tools rather than a separate branch of mathematics, uses of artificial intelligence are naturally included within experimental mathematics (Big Think 2026, Tao 2026).
When a generated result has an independently checkable certificate, exact computation can make this cycle especially effective. A proposed antiderivative can be tested by differentiation, and a proposed counterexample can be checked directly against the relevant definitions. Open-ended search can generate large numbers of candidate identities, patterns, or concepts, but selecting questions and assessing mathematical significance remain distinct from generating candidates (Wolfram 2026).
Tao further contrasts the breadth of this approach with the depth of expert mathematical research. A human researcher typically concentrates on a small number of difficult problems, while automated systems can survey a large collection of problems and may solve a modest fraction of them. Such a count need not include the hardest or most desired problems, so the number of problems solved is not by itself a measure of mathematical significance. In this view, broad automated search complements rather than replaces the depth supplied by human experts (Big Think 2026).
