History of number theory

Most areas of mathematics go from inception to maturity within at most a century. But in number theory there are questions that were formulated more than 2000 years ago (such as whether any odd perfect numbers exist) that have still not been answered. Of the principles that have been established in number theory, a great many were first revealed by explicit experiments. From its inception in classical times, through its development in the 1600s to 1800s, number theory was largely separate from other fields of mathematics. But starting at the end of the 1800s, increasing connections were found to other areas of both continuous and discrete mathematics. And through these connections, sophisticated proofs of such results as Fermat's Last Theorem—open for 350 years—have been constructed. Long considered a rather esoteric branch of mathematics, number theory has in recent years grown in practical importance through its use in areas such as coding theory, cryptography and statistical mechanics. Properties of numbers and certain elementary aspects of number theory have also always played a central role in amateur and recreational mathematics. And as this chapter indicates, number theory can also be used to provide many examples of the basic phenomena discussed in this book.