# Introduction to Number Theory

## Life without Fractions!

Imagine mathematics without fractions or decimals.

We only have integers, which are whole numbers, their negatives, and zero:

This is **Number theory**, a branch of mathematics that explores the properties and relationships of integers.

The cool part is that anyone can join in, it is mostly basic operations like multiplication and addition!

And along the way we will discover many interesting concepts, surprising relationships, and get to rub shoulders with great thinkers of the ages.

## Prime Numbers

Prime numbers play a big part in Number Theory.

The idea that integers above one are either **prime** or the **result of multiplying primes** is really interesting:

And prime factorization is finding which prime numbers multiply together to make a number.

This results in a **unique** representation for every positive integer greater than one.

## Divisibility and Factors

In number theory division is allowed, but we are only interested in integer answers including any remainders.

A central concept is divisibility: can one number be evenly divided by another without any remainder?

This leads to factors: the integers we can multiply together to get our chosen number.

## Modular Arithmetic and Congruences

Modular arithmetic focuses on the remainders when dividing one integer by another.

It is based on the concept of congruence, which happens when numbers have the **same remainder** after being divided by a specific integer.

## Pythagorean Triples and Diophantine Equations

Another interesting aspect of number theory is the study of Pythagorean triples, which are sets of three positive integers that satisfy the Pythagorean theorem a^{2} + b^{2} = c^{2} .

### Example: 3, 4 and 5

a^{2} + b^{2} = c^{2}

3^{2} + 4^{2} = 5^{2}

9 + 16 = 25

They are an example of Diophantine equations (named after the ancient Greek mathematician Diophantus) that explores how to find integer solutions to equations like **a ^{2} + b^{2} = c^{2}**,

**ax + by = c**, etc.

## Cryptography

Cryptography is all about secret messages: how we send messages to people **only they can understand**.

Modern day cryptography is based on the idea that multiplying primes together is much easier than figuring out which primes were multiplied to form a number.

RSA cryptography relies on this concept.

## In Conclusion

Number Theory is a rich and varied field with many important applications.