Generalized Pell Equation and Primes

  • Thread starter ramsey2879
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  • #1
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I have a conjecture that the equation X^2 - 2Y^2 = P has solutions in odd integers if P is a prime of the form 8*N+1. I know of a paper that requires one to find Q such that Q^2 = 2 mod P inorder to solve these equations using continued fractions. To get to first base in proving my conjecture, is there a proof that 2 is a quadratic residue of P where P is a prime of the form 8*N+1?
 

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  • #2
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When p is of the form 8k +-1, then X^2 ==2 Mod (p) is always the case! The theory of quadratic residues was developed by Gauss.
 
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  • #3
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Thanks. Also it is to be noted that I stated my conjecture wrongly. I should have wrote X^2 - 2Y^2 = - P has solutions in odd integers where P is a prime of the form 8*n +1. It is impossible for the other way around to be correct.
 
  • #4
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If p is for the form 4k+1, which satisfies 8k+1, then there is a solution to X^2==-1, Mod P. So for p==1 Mod 8, there is a solution to X^2 ==-2 Mod p.
 
  • #5
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Ramsey 2879: Thanks. Also it is to be noted that I stated my conjecture wrongly. I should have wrote X^2 - 2Y^2 = - P has solutions in odd integers where P is a prime of the form 8*n +1. It is impossible for the other way around to be correct.

Well, there is a question here: 5^2-2(2)^2 = 25-8 = 17,

But: 17 = 2(3^2)-(1)^2 = 18-1.

So you' ll have to look at that again.
 
  • #6
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ramsey2879 said:
My conjecture as corrected stands
Note that for odd integers, X^2- 2Y^2 always equals 7 mod 8. Sorry for the miscommunication.
 
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  • #7
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All odd squaes are congruent to 1 Mod 8. (2x+1)^2 = 4x^2+4x+1 =
4x(x+1) +1 ==1 Mod 8.
 

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