Solvability of the Pell-like equation $x^{2}-dy^{2} = k$

Question

I found some resources that talk about algorithms for finding fundamental solutions to the Pell-like equation $$ x^{2} - dy^{2} = k $$ for $d \in \mathbb{N}$ and $k \in \mathbb{Z}$. However, I'm struggling to find results (if there are any) that will tell me if the above equation is solvable given certain $d,k$. A quick search on Math.SE yields a bunch of questions about specific Pell-like equations.... Does anyone have any good resources/papers related to this question? Any help is greatly appreciated.

Answer 1

You have not said anything about the size of $k.$ So: if you can find a (even) number $\beta$ such that $$ \beta^2 \equiv 4d \pmod {4k}, $$ so that $$ \beta^2 = 4d + 4kt $$ for integer $t.$ So, the discriminant of binary form $$ f(x,y) = k y^2 + \beta xy -t y^2 $$ is $4d.$ There is a process for reducing an indefinite binary form and seeing what class it is in. If $f$ is equivalent to $x^2 - d y^2,$ then $k$ is (primitively) represented by $x^2 - d y^2.$ This does require knowing how to find the complete cycle of a reduced indefinite form. I got the whole business from BUELL

added: let us take notation $$ \langle a, b, c \rangle $$ to refer to the quadratic form $$ f(x,y) = a x^2 + b xy + c y^2 $$ The discriminant is $\Delta = b^2 - 4ac.$ When this is positive but not a square, the form is indefinite. Buell gives the original definition of Gauss and Lagrange for when such a form is reduced.

Proposition: the indefinite (integer coefficient) form $ \langle a, b, c \rangle $ is reduced if and only if both $$ ac < 0 \; \; \; \; \; \mbox{AND} \; \; \; \; \; \; b > |a+c| $$ I know of just one book where this is printed, by Franz Lemmermeyer, who is active on this site and MO. I should give an example, most people haven't seen the cycles of reduced forms:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 5 164 15

  0  form              5         164          15  delta     10
  1  form             15         136        -135


           0          -1
           1          10

To Return  
          10           1
          -1           0

0  form   15 136 -135   delta  -1
1  form   -135 134 16   delta  9
2  form   16 154 -45   delta  -3
3  form   -45 116 73   delta  1
4  form   73 30 -88   delta  -1
5  form   -88 146 15   delta  10
6  form   15 154 -48   delta  -3
7  form   -48 134 45   delta  3
8  form   45 136 -45   delta  -3
9  form   -45 134 48   delta  3
10  form   48 154 -15   delta  -10
11  form   -15 146 88   delta  1
12  form   88 30 -73   delta  -1
13  form   -73 116 45   delta  3
14  form   45 154 -16   delta  -9
15  form   -16 134 135   delta  1
16  form   135 136 -15   delta  -9
17  form   -15 134 144   delta  1
18  form   144 154 -5   delta  -31
19  form   -5 156 113   delta  1
20  form   113 70 -48   delta  -2
21  form   -48 122 61   delta  2
22  form   61 122 -48   delta  -2     ambiguous  
23  form   -48 70 113   delta  1
24  form   113 156 -5   delta  -31
25  form   -5 154 144   delta  1
26  form   144 134 -15   delta  -9
27  form   -15 136 135   delta  1          -1 composed with form zero  
28  form   135 134 -16   delta  -9
29  form   -16 154 45   delta  3
30  form   45 116 -73   delta  -1
31  form   -73 30 88   delta  1
32  form   88 146 -15   delta  -10
33  form   -15 154 48   delta  3
34  form   48 134 -45   delta  -3
35  form   -45 136 45   delta  3
36  form   45 134 -48   delta  -3
37  form   -48 154 15   delta  10
38  form   15 146 -88   delta  -1
39  form   -88 30 73   delta  1
40  form   73 116 -45   delta  -3
41  form   -45 154 16   delta  9
42  form   16 134 -135   delta  -1
43  form   -135 136 15   delta  9
44  form   15 134 -144   delta  -1
45  form   -144 154 5   delta  31
46  form   5 156 -113   delta  -1
47  form   -113 70 48   delta  2
48  form   48 122 -61   delta  -2
49  form   -61 122 48   delta  2     ambiguous  
50  form   48 70 -113   delta  -1
51  form   -113 156 5   delta  31
52  form   5 154 -144   delta  -1
53  form   -144 134 15   delta  9
54  form   15 136 -135


  form   15 x^2  + 136 x y  -135 y^2 

minimum was   5rep   x = -657408301   y = -728220430 disc 26596 dSqrt 163  M_Ratio  118.0844
Automorph, written on right of Gram matrix:  
-3412169357059562966992118401  -34017372299466365140228887000
-3779708033274040571136543000  -37681522192077530811963441601
=========================================

Answer 2

You have to pick the values of your variables but it's not hard to see what they need to be to yield integers once $solutions$ are seen.

$$x=\pm\sqrt{k+dy^2}\quad \land \quad y=\frac{\sqrt{x^2-k}}{\sqrt{d}}$$ For example $$x=\sqrt{2+2(7)^2}=10\quad\land\quad y=\frac{\sqrt{5^2-9}}{\sqrt{4}}=2$$