# Lecture 17

### Powers congruent to $1$

#### Theorem

Let $a$ and $m$ be coprime integers. Then there is some positive $n$ such that $a^n\equiv 1\pmod{m}.$

#### Proof

There are only $m$ different residues modulo $m$, so some two of the sequence $1,a,a^2,a^3,a^4,\ldots,a^m$ must be congruent modulo $m$ (they can’t all be different).

Let’s say that $a^i\equiv a^j\pmod{m}$, with $i.

But $a$ is invertible modulo $m$, and so $(a^{-1})^ia^i\equiv (a^{-1})^ia^j\pmod{m},$ which gives that $a^{j-i}\equiv 1\pmod{m}.$

That proof is a little bit nonconstructive: it tells us it exists, but doesn’t give very much help looking for it.

It turns out that that we can get an explicit result. First we’ll do a relatively easy case, valid when the modulus is prime.

#### Theorem

[Fermat’s Little Theorem] Let $p$ be prime, and let $a$ be an integer coprime to $p$. Then $a^{p-1}\equiv 1\pmod{p}.$

Before we prove it, we’ll talk a while longer about invertible elements and multiplication modulo a prime.

Let’s start with an example, and consider the seven integers $0,3,6,9,12,15,18.$ Regarded modulo 7, each is congruent to something different: \begin{aligned} {0}&{\equiv 0} & {18}&{\equiv 4} \\ {15}&{\equiv 1} & {12}&{\equiv 5} \\ {9}&{\equiv 2} & {6}&{\equiv 6} \\ {3}&{\equiv 3} &&\end{aligned} Can we explain this systematically?

It comes down to the fact that $3$ is invertible modulo $7$ (with inverse $5$, as $3\times 5\equiv{1}\pmod{7}$).

By multiplying congruences, $3\times 5\times a\equiv a\pmod{7}$ so if we want to solve $3x\equiv a\pmod{7}$, we simply take $x\equiv 5a\pmod{7}$.

So as there are seven numbers in the list, and one is congruent to each possible residue $0,1,\ldots,6$ (modulo 7), they’re all different.

This is true in general, for the same reason: if $a$ is coprime to $m$, then the integers $0,a,2a,\ldots,(m-1)a$ contain each of the $m$ residues (and so exactly once each, because there’s $m$ of them).

#### Proof

Consider the product $a\cdot(2a)\cdot(3a)\cdot\cdots\cdot((p-1)a),$ regarded up to congruence modulo $p$.

One way of thinking about it is that it’s $(p-1)!$ but with every term multiplied by an $a$, so is congruent to $a^{p-1}(p-1)!$.

Another is that, since the product contains a copy of every nonzero residue modulo $p$, it is congruent to $(p-1)!$.

But, putting these observations together, we discover that $a^{p-1}(p-1)! \equiv (p-1)!\pmod{p}.$ But all the residues from $1$ to $p-1$ are invertible, and the product of invertible residues is invertible, so $(p-1)!$ is invertible. Multiplying both sides by $(p-1)!^{-1}$ leaves us with $a^{p-1}\equiv1\pmod{p},$ exactly as promised.

Fermat’s Little Theorem should not be confused with Fermat’s Last Theorem. The latter says there are no solutions in positive integers to $a^n+b^n=c^n$ with $n\geq 3$, and was much, much harder to prove.

In the proof of Fermat’s Little Theorem, we multiplied one representative of each invertible residue class together. It turns out we can prove a substantially more general theorem, but it’s a little more complicated. First we need a definition:
Definition: Euler’s function (sometimes known as the totient function) $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ is defined by taking $\varphi(n)$ to be the number of integers from $1$ to $n$ which are coprime to $n$.

For example, $\varphi(p) = p-1$ if $p$ is prime, since every number from $1$ to $p-1$ is coprime to $p$ (and $p$ isn’t coprime to $p$).

For another example, $\varphi(6) = 2$, since $1$ and $5$ are the only numbers between $1$ and $6$ which are coprime to $6$.

Using this concept, we can generalise Fermat’s Little Theorem considerably:

#### Theorem

[Fermat-Euler Theorem] Let $a$ and $n$ be integers with $\gcd(a,n)=1$. Then $a^{\varphi(n)}\equiv 1\pmod{n}.$

#### Proof

The proof is exactly the same as Fermat’s Little Theorem, but instead of working with all the integers $1,2,\ldots,n-1$, we just consider those that are invertible modulo $n$: let’s write these as $x_1,x_2,\ldots,x_{\varphi(n)}$.

If $a$ is invertible, then $ax_1,\ldots,ax_{\varphi(n)}$ are all invertible too, and any invertible residue is of this form: $b$ can be written as $a(a^{-1}b)$. Hence $ax_1,ax_2,\ldots,ax_{\varphi(n)}$ are congruent to $x_1,x_2,\ldots,x_{\varphi(n)}$ in some order.

Hence if we consider the products of these we have \begin{aligned} & x_1x_2\cdots x_{\varphi(n)}\\ \equiv& (ax_1)(ax_2)\cdots(ax_{\varphi(n)})\\ \equiv& a^{\varphi(n)}x_1x_2\cdots x_{\varphi(n)}\pmod{n}\end{aligned} Since all the elements $x_1,x_2,\ldots,x_{\varphi(n)}$ are invertible, we can cancel them out to get $a^{\varphi(n)}\equiv 1\pmod{n}$.