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Will Jagy
Will Jagy
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$$ 106 u^2 + 197 uv + 92 v^2 \; , \; \; 92 u^2 -13 uv + v^2 \; , \; \; u^2 + 15 uv + 106 v^2 \; , \; \; $$$$ 106 u^2 + 197 uv + 92 v^2 \; , \; \; 92 u^2 -13 uv + v^2 \; , \; \; u^2 + 15 uv + 106 v^2 \; . $$

$$ 124 u^2 + 179 uv + 65 v^2 \; , \; \; 65 u^2 -49 uv + 10 v^2 \; , \; \; 10 u^2 + 69 uv + 124 v^2 \; , \; \; $$$$ 124 u^2 + 179 uv + 65 v^2 \; , \; \; 65 u^2 -49 uv + 10 v^2 \; , \; \; 10 u^2 + 69 uv + 124 v^2 \; . $$

$$ 130 u^2 + 159 uv + 49 v^2 \; , \; \; 49 u^2 -61 uv + 20 v^2 \; , \; \; 20 u^2 + 101 uv + 130 v^2 \; . \; \; $$$$ 130 u^2 + 159 uv + 49 v^2 \; , \; \; 49 u^2 -61 uv + 20 v^2 \; , \; \; 20 u^2 + 101 uv + 130 v^2 \; . $$

$$ 106 u^2 + 197 uv + 92 v^2 \; , \; \; 92 u^2 -13 uv + v^2 \; , \; \; u^2 + 15 uv + 106 v^2 \; , \; \; $$

$$ 124 u^2 + 179 uv + 65 v^2 \; , \; \; 65 u^2 -49 uv + 10 v^2 \; , \; \; 10 u^2 + 69 uv + 124 v^2 \; , \; \; $$

$$ 130 u^2 + 159 uv + 49 v^2 \; , \; \; 49 u^2 -61 uv + 20 v^2 \; , \; \; 20 u^2 + 101 uv + 130 v^2 \; . \; \; $$

$$ 106 u^2 + 197 uv + 92 v^2 \; , \; \; 92 u^2 -13 uv + v^2 \; , \; \; u^2 + 15 uv + 106 v^2 \; . $$

$$ 124 u^2 + 179 uv + 65 v^2 \; , \; \; 65 u^2 -49 uv + 10 v^2 \; , \; \; 10 u^2 + 69 uv + 124 v^2 \; . $$

$$ 130 u^2 + 159 uv + 49 v^2 \; , \; \; 49 u^2 -61 uv + 20 v^2 \; , \; \; 20 u^2 + 101 uv + 130 v^2 \; . $$

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Will Jagy
Will Jagy
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To illustrate finding all primitive integer solutions, let us take $(p,q,r) = (106,92,1)$ In this case, all $x,y,z$ will be positive, and we may simply put them in decreasing order. There are three distinct parametrizations, each component a quadratic form in (coprime) integer variables $u,v,$ just like Pythagorean Triples...

$$ 106 u^2 + 197 uv + 92 v^2 \; , \; \; 92 u^2 -13 uv + v^2 \; , \; \; u^2 + 15 uv + 106 v^2 \; , \; \; $$

$$ 124 u^2 + 179 uv + 65 v^2 \; , \; \; 65 u^2 -49 uv + 10 v^2 \; , \; \; 10 u^2 + 69 uv + 124 v^2 \; , \; \; $$

$$ 130 u^2 + 159 uv + 49 v^2 \; , \; \; 49 u^2 -61 uv + 20 v^2 \; , \; \; 20 u^2 + 101 uv + 130 v^2 \; . \; \; $$

Take a pair $(u,v)$ and produce the triples given by each recipe above. If any triple comes out nonprimitive (some gcd > 1) just discard that. The primitive version will be constructed by one of the other recipes. Oh, and put in decreasing order. Call that triple $(x,y,z).$ Below, the triple in brackets is the first quadratic form in that recipe, such as <106,197,92>.

    x          y         z       first quad. form     u  v
   106        92         1      < 106, 197, 92 >      1  0    
   124        65        10      < 124, 179, 65 >      1  0    
   130        49        20      < 130, 159, 49 >      1  0    
   338       251         8      < 130, 159, 49 >      1  1    
   368       203        26      < 124, 179, 65 >      1  1    
   395       122        80      < 106, 197, 92 >      1  1    
   887       412        94      < 130, 159, 49 >      2  1    
   919       302       172      < 124, 179, 65 >      2  1    
  1333      1246         8      < 124, 179, 65 >      1  3    
  1493      1048        46      < 130, 159, 49 >      1  3    
  1525      1000        62      < 106, 197, 92 >      1  3    
  1637       790       160      < 106, 197, 92 >      3  1    
  1696       613       278      < 130, 159, 49 >      3  1    
  1718       448       421      < 124, 179, 65 >      3  1    
  1915      1856        10      < 130, 159, 49 >      2  3    
  2155      1570        56      < 124, 179, 65 >      2  3    
  2270      1880        29      < 124, 179, 65 >      1  4    
  2320      1306       155      < 130, 159, 49 >      3  2    
  2434      1048       299      < 106, 197, 92 >      2  3    
  2450      1000       331      < 124, 179, 65 >      3  2    
  2504      1550       125      < 130, 159, 49 >      1  4    
  2504       754       523      < 106, 197, 92 >      3  2