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Subject: "Pellian Equations"     Previous Topic | Next Topic
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neat_maths
Member since Aug-22-03
Jan-07-06, 11:01 PM (EST)
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"Pellian Equations"
 
   I understand the solutions to Pellian equations of the type

x^2 - n * y^2 = c lie in continued fractions.

Can anyone show me the steps to solve
x^2 - 11 * y^2 = 23
for the smallest positive integer values of x and y
or
x^2 - 11 * y^2 = -23
or
x^2 = 23 - 11 * y^2

take care


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  Subject     Author     Message Date     ID  
Pellian Equations neat_maths Jan-07-06 TOP
  RE: Pellian Equations alexb Jan-08-06 1
     RE: Pellian Equations alexb Jan-08-06 2
         RE: Pellian Equations neat_maths Jan-09-06 4
     RE: Pellian Equations neat_maths Jan-09-06 3
         RE: Pellian Equations alexb Jan-09-06 5
             RE: Pellian Equations neat_maths Jan-10-06 6
                 RE: Pellian Equations alexb Jan-10-06 8

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alexb
Charter Member
1739 posts
Jan-08-06, 12:51 PM (EST)
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1. "RE: Pellian Equations"
In response to message #0
 
   >x^2 - 11 * y^2 = 23

This equation has no integer solutions, since for no x we have

x2 = 2 (mod 11).

>x^2 - 11 * y^2 = -23

This one does have integer solutions.

The best introduction into the matter is Chapter 20 in Dorrie's 100 Great Problems of Elementary Mathematics.

The use of continued fractions is based on the fact that the surds are easily representable as such.


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alexb
Charter Member
1739 posts
Jan-08-06, 12:53 PM (EST)
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2. "RE: Pellian Equations"
In response to message #1
 
   Or check

https://mathworld.wolfram.com/PellEquation.html


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neat_maths
Member since Aug-22-03
Jan-09-06, 07:38 AM (EST)
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4. "RE: Pellian Equations"
In response to message #2
 
   I checked this out first and it appears to indicate that
x^2 - 11*y^2 = 23 is solvable but I haven't been able to do so.

take care


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neat_maths
Member since Aug-22-03
Jan-09-06, 07:38 AM (EST)
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3. "RE: Pellian Equations"
In response to message #1
 
   >>x^2 - 11 * y^2 = 23
>
>This equation has no integer solutions, since for no x we
>have
>
>x2 = 2 (mod 11).

oops I think 23 is 1 (mod 11).

take care


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alexb
Charter Member
1739 posts
Jan-09-06, 07:39 AM (EST)
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5. "RE: Pellian Equations"
In response to message #3
 
   >oops I think 23 is 1 (mod 11).

Oops.


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neat_maths
Member since Aug-22-03
Jan-10-06, 09:31 AM (EST)
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6. "RE: Pellian Equations"
In response to message #5
 
   Actually it is not solvable because x^2 = 11 is impossible mod(23)

take care


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alexb
Charter Member
1739 posts
Jan-10-06, 09:33 AM (EST)
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8. "RE: Pellian Equations"
In response to message #6
 
   >Actually it is not solvable because x^2 = 11 is impossible
>mod(23)
>
Good. Who needs continued fractions?


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