Introduction

``The object of numerical computation is theoretical advance.'' - Bryan Birch describing A.O.L. Atkin.

Much progress in number theory has been driven by attempts to prove conjectures. It's reasonably easy to play around with integers, see a pattern, and make a conjecture. Frequently proving the conjecture is extremely difficult. In this direction, computers help us to

• find more conjectures
• disprove conjectures
• increase our confidence in a conjecture

They also frequently help to solve a specific problem. For example, the following problem would be hopelessly tedious by hand. Here's an example of such a problem:

Find all integer $n<100$ that are the area of a right triangle with integer side lengths.¹

This problem can be solved by a combination of very deep theorems, a few big computer computations, and a little luck.