Introduction

Since we can add, subtract, multiply, divide even... what would be missing? Powering! I'm not talking about some power fantasy here, but rather introduce some really really important theorems. Fermat little's theorem proves useful in a great deal of situation, and is along with Euler's theorem a piece of arithmetic you need to know. Arguably the most canonical example of using these is the RSA cryptosystem, whose decryption step is built around Euler's theorem.

Fermat's Little Theorem

Since we want to talk about powers, let's look at powers. And because I like 7, I made a table of all the powers of all the integers modulo 7.

On the last row, there is a clear pattern emerging, what's going on??? Hm, let's try again modulo 5 this time.

Huh, again?! Clearly, there is something going on... Sage confirms this!

p, itworks = 1, True
for _ in range(100):
    p = next_prime(p)
    Fp = GF(p) # Finite Field of size p
    itworks &= all(Fp(x)^(p-1) == 1 for x in range(1,p))

print(itworks)
# True

Wilson's Theorem

A positive integer $n > 1$is a prime if and only if:

Euler's Theorem

Let $n \in \mathbb{Z}^{+}$ and $a \in \mathbb{Z}$ s.t. $gcd(a, n) = 1$, then:

Fermat's Little Theorem

Let $p$be a prime and $a \in \mathbb{Z}$, then:

or equivalently:

Reference

1. Wilson's Theorem - Brilliant

2. Euler's Theorem - Brilliant

3. Fermat's Little Theorem - Brilliant