Number theory (or arithmetic) 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).

Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory". (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

Andrew James Granville (born 1962) is a British mathematician, working in the field of number theory.

He has been a faculty member at the Université de Montréal since 2002. Before moving to Montreal he was a mathematics professor at University of Georgia (UGA) from 1991 until 2002. He was a section speaker in the 1994 International Congress of Mathematicians together with Carl Pomerance from UGA.

Granville received his Bachelor of Arts (Honours) (1983) and his Certificate of Advanced Studies (Distinction) (1984) from Trinity College, Cambridge University. He received his Ph.D. from Queen's University in 1987 and was inducted into the Royal Society of Canada in 2006.

Granville's work is mainly in number theory, in particular analytic number theory. Along with Carl Pomerance and W. R. (Red) Alford he proved the infinitude of Carmichael numbers in 1994. This proof was based on a conjecture given by Paul Erdős.

In 2008, he won the Chauvenet Prize from the Mathematical Association of America for his paper "It is easy to determine whether a given integer is prime".