Short answers, pulled from the story.
Number theory is a branch of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers, rational numbers, and generalizations of the integers such as algebraic integers. It is one of the oldest branches of mathematics alongside geometry.
Carl Friedrich Gauss, the German mathematician who lived from 1777 to 1855, called number theory the queen of mathematics. He said, "Mathematics is the queen of the sciences and number theory is the queen of mathematics." Gauss wrote Disquisitiones Arithmeticae in 1801, which set the agenda for much of 19th-century number theory.
The main subdivisions of number theory are elementary number theory, analytic number theory, and algebraic number theory. Elementary number theory uses basic arithmetic methods, analytic number theory relies on complex analysis and calculus, and algebraic number theory studies algebraic number fields. Further branches include probabilistic, combinatorial, computational, and applied number theory, along with Diophantine geometry.
Number theory underpins public-key cryptography algorithms such as the RSA cryptosystem, which became prominent in the 1970s. RSA relies on the difficulty of factoring large composite numbers into their prime factors. Fast algorithms for testing primality are known, but no truly fast algorithm for factoring has been found.
The Riemann hypothesis concerns the zeta function, which Bernhard Riemann extended to a complex variable in 1859. He conjectured that all nontrivial zeros of the function occur where the real part equals one half. The hypothesis remains unsolved, and a solution would carry direct consequences for understanding the distribution of primes.
Number theory deals with statements that are simple to understand but very difficult to solve. Fermat's Last Theorem was proved 358 years after its original formulation, and Goldbach's conjecture has remained unsolved since the 18th century. Many results can be stated to a layperson, yet their proofs draw on an unusually broad range of mathematical tools.