How can computers assist mathematicians with calculations and proofs?

Computers serve as powerful tools in mathematics, assisting with both the discovery of theorems and the completion of complex proofs.

Discovery and Computation

Mathematicians use computers to perform extensive calculations, which can help identify patterns and suggest potential theorems, such as the prime number theorem. In applied mathematics, computers are essential for solving differential equations by providing approximate solutions for problems that are difficult to solve exactly. Additionally, symbolic software packages like Mathematica and Maple allow for exact solutions through algebraic manipulation.12

Proof Assistance

Computers assist in completing proofs that require the examination of a vast number of cases. A notable example is the four-colour theorem, where a computer not only verified cases but also helped determine the necessary steps for the proof. While computers are invaluable for these tasks, they generally cannot be used to discover proofs independently, except in restricted areas like elementary Euclidean geometry. As mathematical research evolves, the integration of computers into the formulation and verification of new theorems is becoming an increasingly significant trend in the field.13