In this blog post I will show you how you can use the tools developed in parts 1, 2 and 3 to solve a problem in number theory.

**Determine all the integer solutions of the equations x**^{2} − 3y^{2} = m for m = −1 and 13.

^{2}− 3y

^{2}= m for m = −1 and 13.

Noting that x^{2} − 3y^{2} = (x – √3y)(x + √3), first we find the fundamental unit, *e = a + b√3*, of the number field Q(√3):

For b = 1, 3b^{2} + 1 = 4 = 2^{2 }and so a = 2. Hence the fundamental unit is e* = 2 +√3*.

First we note that the **norm** of an element in Q(√d) is:

**N(a + b√d) = a ^{2} − db^{2}**

We will use the following three facts:

- N(x + √3y) = x
^{2}− 3y^{2}= m. - If we have a unit
*u,*then N(*a*) = N(*ua*) for some*a*in Q(√*d*), as N(*u*) = 1 and N(*ua*) = N(*u*)N(*a*) (i.e. norm is multiplicative). - In the last post, we saw how the units in the ring of algebraic integers of L is of the form:

Thus, multiplying x + √3y by a unit is equal to multiplying x + √3y by a power of the fundamental unit *e*.

So how do we solve this equation? First, by fact 1 we have to do find the element of Q(√3) that has norm +*m* or* -m* (there will be one such element for each, up to multiplication by a unit). Using fact 1 and 2, we deduce all the solutions to the equation are given by** (x, y)** such that:

**N(e) = – 1, N(x + √3y) = -m:****e**^{n}(x + √3y) for*n*a natural number.**N(e) = – 1, N(x + √3y) = m:****e**^{2n}(x + √3y) for*n*a natural number.**N(e) = 1, N(x + √3y) = m: e**^{n}(x + √3y) for*n*a natural number.

## m = -1

Noting that for any element *a*,|N(a)| = 1 if and only if *a* is a unit. Hence as the solutions are given by (x, y) where N(x + √3y) = -1, we must have that x + √3y is a unit. But,

so all units have norm 1, and thus there are NO integer solutions to this equation.

## m = 13

We need to find (x, y) such that N(x + √3y) = x^{2} − 3y^{2} = 13 (because N(*e*) = 1).

- When y = 1, x
^{2}= 14, so x is not an integer. - When y = 2, x
^{2}= 25, so x = 5.

Hence we get that all solutions are of the form:

**e ^{n}(5 + 2√3), for n a natural number and e = 2 +√3, fundamental unit**

This is the last post of this series! Next I’ll be delving into Galois Theory. M x