Number theory, ancient and modern John Coates Summary: Number theory, ancient and modern John Coates 1 Introduction Number theory is the branch of mathematics concerned with the study of the mys- terious and hidden properties of the most basic mathematical objects, namely the integers Z = {0, ±1, ±2, · · ·}, and the rational numbers Q = {m/n : m, n Z and n = 0}. It is the oldest part of mathematics, having its origins somewhere in Asia long before Greek mathematics (e.g. triples of integers, which are the side lengths of right-angled triangles, occur in Babylonian Cuneiform texts dating from 1900- 1600 BC, and in Indian sutras dating from about 800 BC). Since the earliest time until the present day, it has been an experimental science. Number theorists look for the appearance of unexpected patterns and laws in numerical data, and then try to formulate general conjectures. Some of the many unproven conjectures are very old, including one we shall discuss, which can be traced back to Arab manuscripts a thousand years ago. The hardest part of number theory is to find proofs of conjectures, or more usually proofs of partial results in support of these conjectures. When proofs have been found in the past, they have nearly always Collections: Mathematics