As I was looking through different proofs to use for my communicating
mathematics post, I came across a proof that I had never even heard of before.
This proof was called Fermat’s Little Theorem, which is not to be confused with
Fermat’s Last Theorem. This proof states the following

Let

*p*be a prime number which does not divide the integer a, then*a*.^{p}-1 = 1(mod p)
True to Fermat’s form, a proof of this
was not provided. Euler was the first to finish and publish the proof. Although
a proof almost identical to Euler’s was found in a manuscript of Leibniz fifty
years prior to that of Euler.

To begin this proof, we will list the
first

*p-1*multiples of*a*, which looks like this:*a, 2a, 3a, … (p-1)a*

After this, assume that we have some

*ra*and*sa*that are the same modulo*p*. This means that we can conclude*r = s(mod p)*

and also the

*p-1*multiples of*a*above are distinct and nonzero. This means that they must be congruent to*1, 2, 3, … ,(p-1)*in some way. The next step of this proof is to multiply each of these congruencies together in order to obtain the following*a*2a*3a*…*(p-1)a = 1*2*3*…*(p-1)(mod p).*

We can further simplify this as the
following

*a*

^{p-1}(p-1)! = (p-1)!(mod p)

Now, the last step of this proof is to
divide each by

*(p-1)!*. By doing this, we can see that we will get*a*

^{p-1}= 1(mod p)

This is exactly what we were trying to
show. Therefore, we have just proved Fermat’s Little Theorem.

Fermat’s Little Theorem is important to
math because

*a*is so easy to calculate, that most primality tests are built using a version of this theorem.^{p-1}
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