I don't see that this need be tied to Pell's equation; I am taking your $b,c$ squarefree for ease. I have also picked the product so that there are no imprimitive forms of this discriminant. In comparison, if i had picked $5x^2 + 11 y^2,$ I would have had $ 2x^2 + 2xy + 28y^2,$ $4 x^2 + 2xy + 14 y^2,$ $8 x^2 + 6xy+8y^2$ to worry about.

With all this, we take $bx^2 + c y^2$ to be in the principal genus, here $17 x^2 + 89 y^2.$ Then the squares it represents (primitively) are squares of numbers that are primitively represented by its square roots in the class group of forms. The squares that are ruled out are those that share factors with the discriminant, $-4bc.$ For example, below we see plenty of even numbers or multiples of $17$ that are primitively represented by either $ 19 x^2 + 16 xy + 83 y^2$ or $ 38 x^2 + 22 xy + 43 y^2,$ but squares of numbers divisible by any of $2,17,89$ cannot be primitively represented by $17 x^2 + 89 y^2.$

I had a link to your inequality question before i went to the grocery store, then the car developed a coolant leak and I had to drop it at the mechanic and walk home. I think the point was that if $x^2$ is represented, then $x$ itself is represented by a different form of the same discriminant. Furthermore your $y,z$ come from Gauss duplication on that other form.

Not my day.

I will need to think some more about your forms outside the principal genus.

```
=====================================================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./classGroup
Absolute value of discriminant?
6052
Discr -6052 = 2^2 * 17 * 89 class number 16
all
6052: < 1, 0, 1513> Square 6052: < 1, 0, 1513>
6052: < 2, 2, 757> Square 6052: < 1, 0, 1513>
6052: < 11, -8, 139> Square 6052: < 34, 34, 53>
6052: < 11, 8, 139> Square 6052: < 34, 34, 53>
6052: < 17, 0, 89> Square 6052: < 1, 0, 1513>
6052: < 19, -16, 83> Square 6052: < 17, 0, 89>
6052: < 19, 16, 83> Square 6052: < 17, 0, 89>
6052: < 22, -14, 71> Square 6052: < 34, 34, 53>
6052: < 22, 14, 71> Square 6052: < 34, 34, 53>
6052: < 29, -26, 58> Square 6052: < 2, 2, 757>
6052: < 29, 26, 58> Square 6052: < 2, 2, 757>
6052: < 34, 34, 53> Square 6052: < 1, 0, 1513>
6052: < 37, -4, 41> Square 6052: < 2, 2, 757>
6052: < 37, 4, 41> Square 6052: < 2, 2, 757>
6052: < 38, -22, 43> Square 6052: < 17, 0, 89>
6052: < 38, 22, 43> Square 6052: < 17, 0, 89>
squares
6052: < 1, 0, 1513>
6052: < 2, 2, 757>
6052: < 17, 0, 89>
6052: < 34, 34, 53>
fourths
6052: < 1, 0, 1513>
Discriminant -6052 h : 16 Squares : 4 Fourths : 1
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./squareprimitivego
===========================================================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primgo
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2
19 16 83
Discriminant 6052
Maximum number represented?
1000
19 = 19
83 = 83
86 = 2 * 43
118 = 2 * 59
127 = 127
191 = 191
206 = 2 * 103
302 = 2 * 151
319 = 11 * 29
323 = 17 * 19
383 = 383
407 = 11 * 37
451 = 11 * 41
478 = 2 * 239
599 = 599
638 = 2 * 11 * 29
647 = 647
671 = 11 * 61
718 = 2 * 359
727 = 727
814 = 2 * 11 * 37
859 = 859
863 = 863
902 = 2 * 11 * 41
919 = 919
967 = 967
982 = 2 * 491
--------------------------------------------------------------------------
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primgo
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2
38 22 43
Discriminant 6052
Maximum number represented?
1000
38 = 2 * 19
43 = 43
59 = 59
103 = 103
151 = 151
166 = 2 * 83
239 = 239
254 = 2 * 127
319 = 11 * 29
359 = 359
382 = 2 * 191
407 = 11 * 37
451 = 11 * 41
491 = 491
563 = 563
638 = 2 * 11 * 29
646 = 2 * 17 * 19
671 = 11 * 61
731 = 17 * 43
739 = 739
766 = 2 * 383
814 = 2 * 11 * 37
883 = 883
902 = 2 * 11 * 41
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$
------------------------------------------------------------------
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./squareprimitivego
Input three coefficients a b c for positive f(x,y)= a x^2 + b x y + c y^2
17 0 89
Discriminant -6052
Maximum number represented?
1000000
361 = 19^2
1849 = 43^2
3481 = 59^2
6889 = 83^2
10609 = 103^2
16129 = 127^2
22801 = 151^2
36481 = 191^2
57121 = 239^2
101761 = 11^2 * 29^2
128881 = 359^2
146689 = 383^2
165649 = 11^2 * 37^2
203401 = 11^2 * 41^2
241081 = 491^2
316969 = 563^2
358801 = 599^2
418609 = 647^2
450241 = 11^2 * 61^2
528529 = 727^2
546121 = 739^2
737881 = 859^2
744769 = 863^2
779689 = 883^2
844561 = 919^2
935089 = 967^2
================================================================
```