Fermat’s Little Theorem implies Wilson’s Theorem

disquisitionesmathematicae — Sat, 02 May 2009 08:44:06 +0000

I came across this novel argument in a book last year. It basically exploits the identity,

Wilson’s theorem states that

The result is true for . So we may assume 2' title='p>2' class='latex' />. We then conveniently consider the above expression for .

Choosing we get

Using Fermat’s little theorem and the fact that ‘p’ is odd we get,

The Pascal’s identity,

then implies,

And since it follows that

Therefore,

And thus,

Counting and the Lagrange identity.

disquisitionesmathematicae — Wed, 29 Apr 2009 18:04:32 +0000

This is a prime example of how a proof for a general argument can be worked out by considering simple cases. The Lagrange identity for says that

We will analyze the case for n=3. It will provide a beautiful outline for a general proof.

The left hand side of the equality (1) will be equal to

It would be helpful indeed to consider the following 3 x 3 matrix for the sum of all the elements of this matrix is equal to the LHS of (2).

In fact, the elements of the matrix are the terms on the RHS in (2).

Each term of the matrix above can thus be represented as

Now consider the first term on the RHS in (1). For n=3, it is

Again, as above, it would be helpful to consider the following matrix.

The elements of this matrix are again the terms on RHS in (3).

We can thus call each of them

The last term in (1) is

This time we use two matrices to organize the terms in (4).

and

And we thus define elements of the first and the second matrix respectively as

where we define as

Now let

Therefore,

where we have used ( is the Kronecker delta)

And so,

Since each is zero the sum

which automatically implies the Lagrange Identity in .