 
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
rightangled 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
