Primitive Pythagorean triples such that 2uv(u^2 – v^2) = x^2 + 3*y^2

 
 
To find Primitive Pythagorean triples such that the product of the legs can be expressed as   x^2 + 3 \, y^2

 
Here are the first few examples,
 

(3, 4, 5)   …………   x^2 \; + \; 3 \, y^2 \; = \; 3\times 4
(x,y) = (3,1), (0,2)

(16, 63, 65)   ……..   x^2 \; + \; 3 \, y^2 \; = \; 16\times 63
(x,y) = (30,6),(24,12),(6,18)

(13, 84, 85)   ……..   x^2 \; + \; 3 \, y^2 \; = \; 13\times 84
(x,y) = (33,1),(30,8),(27,11),(18,16),(15,17),(3,19)

(133, 156, 205)   …..   x^2 \; + \; 3 \, y^2 \; = \; 133\times 156
(x,y) = (144,2),(141,17),(135,29),(129,37),(120,46),(111,53),(96,62),(75,71),(69,73),(45,79),(24,82),(9,83)

 

Find few more examples
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to Primitive Pythagorean triples such that 2uv(u^2 – v^2) = x^2 + 3*y^2

  1. paul says:

    Here are a few more, format is {a, b, c, ab}, {{x, y}}.

    {3,4,5,12} , {{3,1}}
    {6,8,10,48} , {{6,2}}
    {9,12,15,108} , {{9,3}}
    {12,16,20,192} , {{12,4}}
    {13,84,85,1092} , {{3,19},{15,17},{18,16},{27,11},{30,8},{33,1}}
    {15,20,25,300} , {{15,5}}
    {16,63,65,1008} , {{6,18},{24,12},{30,6}}
    {18,24,30,432} , {{18,6}}
    {21,28,35,588} , {{9,13},{15,11},{21,7},{24,2}}
    {24,32,40,768} , {{24,8}}
    {26,168,170,4368} , {{6,38},{30,34},{36,32},{54,22},{60,16},{66,2}}
    {27,36,45,972} , {{27,9}}
    {27,364,365,9828} , {{9,57},{45,51},{54,48},{81,33},{90,24},{99,3}}
    {30,40,50,1200} , {{30,10}}
    {32,126,130,4032} , {{12,36},{48,24},{60,12}}
    {33,44,55,1452} , {{33,11}}
    {36,48,60,1728} , {{36,12}}
    {37,684,685,25308} , {{51,87},{75,81},{84,78},{105,69},{156,18},{159,3}}
    {39,52,65,2028} , {{21,23},{24,22},{39,13},{45,1}}
    {39,252,255,9828} , {{9,57},{45,51},{54,48},{81,33},{90,24},{99,3}}
    {42,56,70,2352} , {{18,26},{30,22},{42,14},{48,4}}
    {45,60,75,2700} , {{45,15}}
    {48,64,80,3072} , {{48,16}}
    {48,189,195,9072} , {{18,54},{72,36},{90,18}}
    {49,1200,1201,58800} , {{90,130},{150,110},{210,70},{240,20}}

    Paul.

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