Number theory is the branch of pure mathematics deals with the properties of numbers in general, and mostly integers, as well as the wider classes of problems that arise from their study. Please see the book Number Theory for a detailed treatment. In elementary number theory, integers are studied without use of techniques from other mathematical fields.

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Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, integer factorizations into prime numbers, investigation of perfect numbers and congruences belong here. Several important discoveries of this field are Fermat's little theorem, Euler's theorem, the Chinese remainder theorem and the law of quadratic reciprocity. Many questions in number theory can be stated in elementary number theoretic terms, but they may require very deep consideration and new approaches outside the realm of elementary number theory to solve.

Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients. An algebraic number field is any finite and therefore algebraic field extension of the rational numbers. These domains contain elements analogous to the integers, the so-called algebraic integers.

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## Elementary Number Theory

In this setting, the familiar features of the integers e. The virtue of the machinery employed — Galois theory, group cohomology, class field theory, group representations and L-functions — is that it allows one to recover that order partly for this new class of numbers.

Analytic number theory is the branch of the number theory that uses methods from mathematical analysis to prove theorems in number theory. Euclid's algorithm. Linear Diophantine Equations.

## Rosen, Elementary Number Theory, 6th Edition | Pearson

The Chinese Remainder Theorem. Primitive roots. Quadratic reciprocity. Arithmetic functions. Diophantine equations 1. Diophantine equations 2. Continued fractions. Periodic continued fractions.

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