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