Waste less time on Facebook — follow Brilliant.
×

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?

Note by John Ashley Capellan
3 years, 4 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} \( \boxed{123} \)

Comments

Sort by:

Top Newest

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

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

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 - 3 years, 4 months ago

Log in to reply

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

Mathh Mathh - 2 years, 11 months ago

Log in to reply

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

Calvin Lin Staff - 3 years, 2 months ago

Log in to reply

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.

Subrata Saha - 3 years, 4 months ago

Log in to reply

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

Usama Khidir - 3 years, 4 months ago

Log in to reply

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

Subrata Saha - 3 years, 4 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...