For a specific problem, there is a finite check to decide whether any solutions exist to $x^2 - a y^2 = b.$ Find the minimum positive integrs $u,v > 0$ such that $u^2 - a v^2 = 1.$ Any solution to your problem, say with $x,y$ positive, creates an infinite sequence of solutions by
$$ (x,y) \mapsto (ux + av y, \; vx + u y). $$ This will increase the entries. In the other direction, that of decreasing one or both entries,
$$ (x,y) \mapsto (ux - av y, \; -vx + u y). $$
Repeating this mapping gets us to a solution with both $x,y> 0$ but one of the entries in the left neighbor nonpositive, either
$$ ux - ay \leq 0 \; \; \; \mbox{OR} \; \; \; -vx + uy \leq 0. $$
If you draw some picture, including the hyperbola $x^2 - a y^2 = b,$ you see how one or the other of $ux \leq ay$ or $uy \leq vx$ gives a bounded arc of the hyperbola. If there are no integer solutions in that arc, there are no solutions at all.
More generally, one may draw the Conway topograph for an indefinite form $a x^2 + bxy + c y^2.$ His "climbing lemma" shows how we need investigate only a finite region of the diagrma to decide whether there is a (primitive) solution to $a x^2 + bxy + c y^2= n.$
$$ x^2 - 2 y^2 = 7 \cdot 17 \cdot 23 $$
jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
Automorphism matrix:
3 4
2 3
Automorphism backwards:
3 -4
-2 3
3^2 - 2 2^2 = 1
x^2 - 2 y^2 = 2737
Sun Mar 12 14:48:31 PDT 2017
x: 53 y: 6 ratio: 8.83333 SEED KEEP +-
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x: 57 y: 16 ratio: 3.5625 SEED KEEP +-
x: 73 y: 36 ratio: 2.02778 SEED KEEP +-
x: 75 y: 38 ratio: 1.97368 SEED BACK ONE STEP 73 , -36
x: 107 y: 66 ratio: 1.62121 SEED BACK ONE STEP 57 , -16
x: 117 y: 74 ratio: 1.58108 SEED BACK ONE STEP 55 , -12
x: 135 y: 88 ratio: 1.53409 SEED BACK ONE STEP 53 , -6
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Sun Mar 12 14:48:51 PDT 2017
x^2 - 2 y^2 = 2737
jagy@phobeusjunior:~$
$$ x^2 - 2 y^2 = - 7 \cdot 17 \cdot 23 $$
jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
Automorphism matrix:
3 4
2 3
Automorphism backwards:
3 -4
-2 3
3^2 - 2 2^2 = 1
x^2 - 2 y^2 = -2737
Sun Mar 12 14:50:52 PDT 2017
x: 1 y: 37 ratio: 0.027027 SEED KEEP +-
x: 25 y: 41 ratio: 0.609756 SEED KEEP +-
x: 31 y: 43 ratio: 0.72093 SEED KEEP +-
x: 41 y: 47 ratio: 0.87234 SEED KEEP +-
x: 65 y: 59 ratio: 1.10169 SEED BACK ONE STEP -41 , 47
x: 79 y: 67 ratio: 1.1791 SEED BACK ONE STEP -31 , 43
x: 89 y: 73 ratio: 1.21918 SEED BACK ONE STEP -25 , 41
x: 145 y: 109 ratio: 1.33028 SEED BACK ONE STEP -1 , 37
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Sun Mar 12 14:51:24 PDT 2017
x^2 - 2 y^2 = -2737
jagy@phobeusjunior:~$