Make {x^2 + 3xy + y^2, x^2 + 5xz + z^2, y^2 + 7yz + z^2} squares

 
 
                                                                            #1                      

 

x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2
x^2 \; + \; 5 \,x \,z \; + \; z^2 \; = \; B^2
y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2

 
Here are the first few solutions:
 

\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}

\{ \, 104, \; 525, \; 120, \; 671, \; 296, \; 855 \, \}
\{ \, 384, \; 493, \; 528, \; 979, \; 1200, \; 1531 \, \}
\{ \, 1736, \; 2397, \; 17112, \; 4609, \; 21080, \; 24201 \, \}

 

                                                                            #2                      

 
x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2
x^2 \; + \; 5 \,x \,z \; + \; z^2 \; = \; B^2
y^2 \; - \; 7 \,y \,z \; + \; z^2 \; = \; C^2

 
 
\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}

\{ \, 3, \; 7, \; 1, \; 11, \; 5, \; 1 \, \}
\{ \, 15, \; 259, \; 13, \; 281, \; 37, 209 \, \}
\{ \, 192, \; 1001, \; 120, \; 1271, \; 408, \; 419 \, \}
\{ \, 315, \; 1079, \; 153, \; 1511, \; 603, \; 179 \, \}
\{ \, 1755, \; 2831, \; 297, \; 5099, \; 2403, \; 1489 \, \}

 

                                                                            #3                      

 
 
x^2 \; - \; 3 \,x \,y \; + \; y^2 \; = \; A^2
x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2
y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2

 
 
\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}
 
the first few solutions:

\{ \, 8, \; 21, \; 40, \; 1, \; 8, \; 89 \, \}
\{ \, 312, \; 817, \; 1664, \; 11, \; 520, \; 3599 \, \}
\{ \, 1368, \; 3901, \; 12920, \; 1039, \; 8968, \; 23129 \, \}

 

                                                                            #4                      

 
x^2 \; + \; 3 \,x \,y \; + \; y^2 \; = \; A^2
x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2
y^2 \; + \; 7 \,y \,z \; + \; z^2 \; = \; C^2

 
\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}

\{ \, 143, \; 27, \; 819, \; 181, \; 325, \; 909 \, \}
\{ \, 285, \; 80, \; 1729, \; 395, \; 779, \; 1991 \, \}
\{ \, 481, \; 1032, \; 5957, \; 1669, \; 4625, \; 8921 \, \}
\{ \, 989, \; 960, \; 8385, \; 2179, \; 5461, \; 11295 \, \}
 
 

                                                                            #5                      

 
x^2 \; - \; 3 \,x \,y \; + \; y^2 \; = \; A^2
x^2 \; - \; 5 \,x \,z \; + \; z^2 \; = \; B^2
y^2 \; - \; 7 \,y \,z \; + \; z^2 \; = \; C^2

 
the first few solutions:

\{ \, x, \; y, \; z, \; A, \; B, \; C \, \}

\{ \, 5, \; 15, \; 1, \; 5, \; 1, \; 11 \, \}
\{ \, 231, \; 680, \; 35, \; 211, \; 119, \; 545 \, \}
\{ \, 741, \; 2135, \; 65, \; 601, \; 559, \; 1895 \, \}
\{ \, 819, \; 2432, \; 143, \; 781, \; 325, \; 1871 \, \}
\{ \, 896, \; 2613, \; 112, \; 779, \; 560, \; 2189 \, \}
\{ \, 1615, \; 4680, \; 171, \; 1355, \; 1121, \; 4041 \, \}

 

Find the next few solutions
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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