# number theory

Number theory is the study of the whole numbers and their properties and relationships. Often, a problem in number theory can be restated in terms of finding the solution, or showing that there is no solution, to a Diophantine equation. A Diophantine equation is one where the coefficients are whole numbers, and where the solution is also constrained to be a whole number. Sometimes, what appears to be a simple Diophantine equation can lead to an elliptic curve.

**Algebraic number theory** is the branch of number theory that is studied without using methods like infinite
series, convergence, and so forth,
taken from analysis. **Analytical number theory** is the branch of number theory that uses
methods taken from analysis, especially complex analysis.

### reference

1. Beiler, Albert H. *Recreations in the Theory of Numbers*.
Mineola, NY: Dover, 1964.