Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (such as rational numbers) or defined as generalizations of the integers (such as, for example, algebraic integers).
An ambitious theory-builder should regard the absence thus far of a Grand Unifed Theory of analytic number theory not as an insult but as a challenge. Both machinery- and problem-motivated mathematicians should note that some of the more exciting recent work in number theory depends critically on contributions from both sides of the stylistic fence.
A hallmark of analytic number theory is the treatment of number-theoretical problems (usually enumerative) by methods often relegated to the domain of "applied mathematics": elementary but clever manipulation of sums and integrals; asymptotic and error analysis; Fourier series and transforms; contour integrals and residues. While there is still good new work to be done along these lines, much contemporary analytic number theory also uses advanced tools from within and outside number theory.
Number theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions.
Here are the steps to follow:
1. Accumulate data, usually numerical, but sometimes more abstract in nature.
2. Examine the data and try to ﬁnd patterns and relationships.
3. Formulate conjectures (i.e., guesses) that explain the patterns and relationships. These are frequently given by formulas.
4. Test your conjectures by collecting additional data and checking whether the new information ﬁts your conjectures.
5. Devise an argument (i.e., a proof) that your conjectures are correct.
Number theory, with its discrete, precise nature, is an ideal topic in which to perform numerical experiments and calculations. Many of these can be done by hand.
Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician ... His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published.
Most mathematics majors first encounter number theory in courses on abstract algebra, for which number theory provides numerous examples of algebraic systems, such as finite groups, rings, and fields. ... Too often, one obtains from number theory texts the impression that each topic has been thoroughly developed. The problems offered in such texts are either solved or unsolvable.
A final reason for learning number theory is that, despite Hardy's (1940) famous but not out-dated claim, it is useful. Its best-known modern application is to the cryptographic systems which allow banks, commercial companies, military establishments and so on to exchange information in securely-encoded form ... Physicists, engineers and computer scientists are also finding that number-theoretic concepts are playing an increasing role in their work.
Number theory has always fascinated amateurs as well as professional mathematicians. In contrast to other branches of mathematics, many of the problems and theorems of number theory can be understood by laypersons, although solutions to the problems and proofs of the theorems often require a sophisticated mathematical background.
Numbers have fascinated civilized man for millennia. The Pythagoreans studied many properties of the natural numbers 1, 2, 3, ... , and the famous theorem of Pythagoras, though geometrical, has a pronounced number-theoretic content. Earlier Babylonian civilizations had noted empirically many so-called Pythagorean triads, such as 3, 4, 5 and 5, 12, 13.