Quadratic Diophantic equation

Question

Hello :) i want to give a answer op the following question:

For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$.

I want to solve the question without a Computer Algebra system. I want to solve it algebraic. I thought about the following:

We can maybe use the quadratic reciprocity, because $x^2-65y^2=p$ is equivalent with $(x-y\sqrt{65})(x+y\sqrt{65})=p$ thus we search for primes which splits in the integers of $\Bbb{Q}(\sqrt{65})$. But how to give a general answer? Can someone help me? Thank you :)

Answer 1

Largely because $65 \equiv 1 \pmod 8,$ the form $x^2 - 65 y^2$ represents exactly the same odd numbers as $x^2 + x y - 16 y^2.$ That is to say, residues mod 5 and mod 13. By the Chinese Remainder Theorem. So, all primes $1,4,9,14,16,29,36,49,51,56,61,64 \pmod {65}.$ For example, $$ 1 \pmod {65}: \; \; \; 131 = 14^2 - 65 \cdot 1^2. $$ $$ 4 \pmod {65}: \; \; \; 199 = 28^2 - 65 \cdot 3^2. $$ $$ 9 \pmod {65}: \; \; \; 139 = 42^2 - 65 \cdot 5^2. $$ $$ 14 \pmod {65}: \; \; \; 79 = 12^2 - 65 \cdot 1^2. $$ $$ 16 \pmod {65}: \; \; \; 211 = 74^2 - 65 \cdot 9^2. $$ $$ 29 \pmod {65}: \; \; \; 29 = 17^2 - 65 \cdot 2^2. $$ $$ 36 \pmod {65}: \; \; \; 101 = 19^2 - 65 \cdot 2^2. $$ $$ 49 \pmod {65}: \; \; \; 179 = 58^2 - 65 \cdot 7^2. $$ $$ 51 \pmod {65}: \; \; \; 181 = 21^2 - 65 \cdot 2^2. $$ $$ 56 \pmod {65}: \; \; \; 251 = 106^2 - 65 \cdot 13^2. $$ $$ 61 \pmod {65}: \; \; \; 61 = 49^2 - 65 \cdot 6^2. $$ $$ 64 \pmod {65}: \; \; \; 389 = 83^2 - 65 \cdot 10^2. $$

EEEDDDIIIITTT: In case anyone is looking. The first thing of importance was that $8^2 - 65 \cdot 1^2 = -1.$ That is, (and this is unusual for composite numbers such as 65), we need pay no attention to $\pm$ signs anywhere. I think what I will do next is display the full Lagrange cycle for the principal and the secondary genus, first for discriminant 65 and then for 260.

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
4 1 -4

  0  form              4           1          -4


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   4 1 -4   delta  -1
1  form   -4 7 1   delta  7
2  form   1 7 -4   delta  -1
3  form   -4 1 4   delta  1
4  form   4 7 -1   delta  -7
5  form   -1 7 4   delta  1
6  form   4 1 -4
minimum was   1rep -1 -1 disc   65 dSqrt 8.0622577483  M_Ratio  4.0625
Automorph, written on right of Gram matrix:  
-113  -128
-128  -145
 Trace:  -258   gcd(a21, a22 - a11, a12) : 32
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
    jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
2 7 -2

  0  form              2           7          -2


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   2 7 -2   delta  -3
1  form   -2 5 5   delta  1
2  form   5 5 -2   delta  -3
3  form   -2 7 2   delta  3
4  form   2 5 -5   delta  -1
5  form   -5 5 2   delta  3
6  form   2 7 -2
minimum was   2rep 1 0 disc   65 dSqrt 8.0622577483  M_Ratio  16.25
Automorph, written on right of Gram matrix:  
-17  -64
-64  -241
 Trace:  -258   gcd(a21, a22 - a11, a12) : 32
=========================================

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
1 0 -65

  0  form              1           0         -65  delta      0
  1  form            -65           0           1  delta      8
  2  form              1          16          -1


          -1          -8
           0          -1

To Return  
          -1           8
           0          -1

0  form   1 16 -1   delta  -16
1  form   -1 16 1   delta  16
2  form   1 16 -1
minimum was   1rep 1 0 disc   260 dSqrt 16.124515497  M_Ratio  260
Automorph, written on right of Gram matrix:  
-1  -16
-16  -257
 Trace:  -258   gcd(a21, a22 - a11, a12) : 16
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
    jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
5 0 -13

  0  form              5           0         -13  delta      0
  1  form            -13           0           5  delta      1
  2  form              5          10          -8


          -1          -1
           0          -1

To Return  
          -1           1
           0          -1

0  form   5 10 -8   delta  -1
1  form   -8 6 7   delta  1
2  form   7 8 -7   delta  -1
3  form   -7 6 8   delta  1
4  form   8 10 -5   delta  -2
5  form   -5 10 8   delta  1
6  form   8 6 -7   delta  -1
7  form   -7 8 7   delta  1
8  form   7 6 -8   delta  -1
9  form   -8 10 5   delta  2
10  form   5 10 -8
minimum was   5rep 1 0 disc   260 dSqrt 16.124515497  M_Ratio  10.4
Automorph, written on right of Gram matrix:  
-49  -128
-80  -209
 Trace:  -258   gcd(a21, a22 - a11, a12) : 16
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 

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