Solving Pell Equation of Norms other than -1 and 1

People, I crossed a Pell-type equation: x^2 - 6y^2 = 3 which it has norm 3. Are there ways to solve this equation without using concepts from Abstract Algebra such as factoring in a number field or what?

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Comments

You can read up on Pell's Equation, to understand how to generate solutions from a base case.

1) Observe that $(5,2)$ is the first non-trivial solution to $x^2 - 6y^2 = 1$ .

2) Observe that

$( a^2 - 6b^2 ) ( c^2 - 6 d^2 ) = a^2c^2 + 36 b^2d^2 - 6 b^2 c^2 - 6a^2d^2= ( ac + 6bd) ^2 - 6 ( bc+ad) ^2.$

As such, we define pair-multiplication as $(a,b) \otimes (c, d) = ( ac + 6 bd , bc + ad)$ .

3) Observe that $(3, 1)$ is a solution to $x^2 - 6y^2 = 3$ .

4) Hence, solutions exist in the form of $(3,1) \otimes ( 5,2)^ n$ , where $n$ is a non-negative integer.
For example, with $n=1$ , we get $(3,1) \otimes (5,2) = ( 15 + 12 , 5 + 6) = (27, 11)$ . We can check that $27^2 - 6 \times 11^2 = 729 - 726 = 3$ .

Followup question: Are there other solutions? (ignore negative values)

Calvin Lin Staff - 6 years, 10 months ago

It's $(ac+6bd,bc+ad)$ . Fix it to avoid confusion.

mathh mathh - 6 years, 5 months ago

You also left out an $a^2$ in Point 2. Cheers, G.

A Former Brilliant Member - 2 years, 10 months ago

Fixed. Thanks!

Calvin Lin Staff - 2 years, 10 months ago

@John Ashley Capellan Can you add what you learnt about Pell's Equation to the Wiki? Thanks!

Calvin Lin Staff - 6 years, 8 months ago

First find the smallest positive solution of x,y. Express it as x+6^0.5y Then find solution of the equation x^2-6y^2=1. Express it as x+6^0.5y Multiply any of these two you get another solution. Hence obtain all solutions.

A Former Brilliant Member - 6 years, 10 months ago

Could you please explain in a bit more detail? Sorry for the trouble.

Usama Khidir - 6 years, 10 months ago

You can know about my solution by searching pell fermat equation in wikipedia

A Former Brilliant Member - 6 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$