# Number Theory

Number theory is often regarded as the entryway to pure mathematics because it is extremely rich in interesting and highly applicable problems, yet requires few prerequisites. Some of the most famous mathematicians ever — like Euler, Fermat, and Gauss — pioneered results in number theory, making the subject what it is today.

The main applications of number theory use concepts developed with modular arithmetic to encrypt or decrypt messages, giving rise to cryptography. Before one can understand cryptography systems, from Caesar Ciphers to the RSA Cryptosystem, they must develop a foundational understanding of operations and properties involved in modular arithmetic.

## Lessons

### Number Orders

A description of number orders in modular arithmetic.

### Reciprocity

A description of quadratic reciprocity in modular arithmetic and an introduction to generalized reciprocity.

### Operations

Properties of addition, multiplication, and generalized operations in different settings.

### Totient Function

Definition and description of Euler's totient (phi) function.