📂Number Theory

Pell's equation

Buildup

$a_{n} : = n^2$ is referred to as a Square Number.

$\displaystyle b_{m} : = {{ m ( m + 1 ) } \over {2}}$ is referred to as a Triangular Number.

Considering if there are numbers that are both square and triangular, $a_{1} =b_{1} = 1$ and $\displaystyle a_{6} = 6 ^2 = 36 = {{ 8 \cdot 9 } \over {2}} = b_{8}$ immediately come to mind. Now, let’s think about the general case where a number is both square and triangular. $$ \begin{align*} & n^2 = {{ m ( m + 1 ) } \over {2}} \\ & \implies 8 n^2 = 4 m ( m + 1 ) \\ & \implies 8 n^2 = ( 2 m + 1 )^2 - 1 \end{align*} $$ If we set $x := 2m + 1$ and $y := 2n$, then $$ 2 y^2 = x^2 - 1 $$ This rephrases the question ‘what is a number that is both square and triangular?’ into finding a natural solution for $x^2 - 2 y^2 = 1$.

Definition ¹

The generalised form of such equations is known as Pell’s Equation, and the following theorem is known.

Theorem

• [1]: For $D \in \mathbb{N}$, which is not a perfect square, $x^2 - D y^2 = 1$ always has a solution.
• [2]: If $( x_{1} , y_{1} )$ is the solution with the smallest value for $x_{1}$, then all solutions $(x_{k} , y_{k})$ can be derived as $x_{k} + y_{k} \sqrt{D} = \left( x_{1} + y_{1} \sqrt{D} \right)^{k}$, where $k , x_{k} , y_{k} \in \mathbb{N}$.

Explanation

As a direct follow-up from square and triangular numbers, the natural solution that satisfies $x^2 - 2 y^2 = 1$ is $3^2 - 2 \cdot 2^2 = 1$, hence $(3,2)$ exists. Since $x = 3 = 2m + 1$ and defined as $y = 2 = 2n$, it exactly matches the simplest case of $n= m =1$. Now, considering the case where $k=2$, $$ x_{2} + y_{2} \sqrt{2} = \left( 3 + 2 \sqrt{2} \right)^2 = 17 + 12 \sqrt{2} $$ In fact, since it is defined as $x = 17 = 2m + 1 $ and $ y = 12 = 2n$, these values become the known $m=8$ and $n=6$.

What stands out in Pell’s equation, despite clearly being a part of number theory, is the use of the irrational number $\sqrt{2}$ in calculations. This extension is also possible for complex numbers. Additionally, since the form of the equation is similar to that of a hyperbolic equation, one can surmise that there has been some discussion related to this.