Pell-type equations with no integer solutions

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 integers $u,v > 0$ 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 diagram to decide whether there is a (primitive) solution to $a x^2 + bxy + c y^2= n.$

$$ x^2 - 2 y^2 = 7 \cdot 17 \cdot 23 $$ enter image description here

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 +- 
x:  55  y:  12 ratio: 4.58333  SEED   KEEP +- 
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
x:  183  y:  124 ratio: 1.47581
x:  213  y:  146 ratio: 1.4589
x:  235  y:  162 ratio: 1.45062
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x:  14156043  y:  10009834 ratio: 1.41421
x:  14707657  y:  10399884 ratio: 1.41421

Sun Mar 12 14:48:51 PDT 2017

 x^2 - 2 y^2 = 2737

$$ x^2 - 2 y^2 = - 7 \cdot 17 \cdot 23 $$ enter image description here

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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x:  14695  y:  10391 ratio: 1.4142
x:  17201  y:  12163 ratio: 1.41421
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x:  9480199  y:  6703513 ratio: 1.41421
x:  10488785  y:  7416691 ratio: 1.41421
x:  12277471  y:  8681483 ratio: 1.41421
x:  16958071  y:  11991167 ratio: 1.41421

Sun Mar 12 14:51:24 PDT 2017

 x^2 - 2 y^2 = -2737

$$ x^2 - 2 y^2 = -15 $$ no integer solutions; prohibited by local conditions. enter image description here


Yes, but it is not easy. Check out John Robertson's book.


Yes. First if there are no solutions in $\mathbb{R}$ or in $\mathbb{Z}_p$ for some prime $p$, then there are no solutions in $\mathbb{Z}$.

This condition is not sufficient in general; there could be a Brauer-Manin obstruction to the Hasse principle. For more about the Brauer-Manin obstruction for integral solutions to quadratic equations, see: https://www.math.u-psud.fr/~colliot/CTXuCompositio2009.pdf