Number theory looks at some classical problems concerning the integers, including the solution of Diophantine equations; the distribution of prime numbers; the theory of congruences; quadratic reciprocity; and the theory of continued fractions. Number theoryThis section is concerned with the integers, and in particular with the solution of classical problems that require integer solutions. It begins by considering some elementary properties of the integers, such as divisibility and greatest common divisors. This leads to a method of solving the linear Diophantine equation ax + by = c, that is, finding solutions to the equation that are integers.Every integer greater than 1 is shown to be a unique product of primes, and some results are obtained concerning the distribution of primes among the integers. In the theory of congruences, methods are developed for solving linear congruences such as ax ≡ b {mod n) and the classical theorems of Fermat and Wilson are obtained. We then consider multiplicative functions: functions f satisfying f(m) x f(n) = f(mn) for relatively prime integers m and n, and in particular Euler’s φ-function, which counts the number of integers in the set { 0, 1, …, (n–1)} that are relatively prime to n. Returning to congruences we consider the solution of quadratic congruences, which leads to Gauss’s law of quadratic reciprocity. Finally, the story of continued fractions is developed and applied as a method of solving further examples of Diophantine equations.