Make {x^2 + y^2 + z^2, x^2 y^2 + x^2 z^2 + y^2 z^2} squares – Part 2

 
 
Find three positive integers   (x, y, z)   to can make the two expressions squares

    x^2 \; + \; y^2 \; + \; z^2
x^2 \, y^2 \; + \; x^2 \, z^2 \; + \; y^2 \, z^2

 

Here’s a parametric solution established by Euler:

x \; = \; (n^4 - 6 \, n^2 + 1) \,(n^2 + 1)
y \; = \; 4 \, n \, (n^2 - 1)^2
z \; = \; (8 \, n^2) \,(n^2 - 1)

x^2 + y^2 + z^2 = (n^2+1)^6
x^2 \, y^2 + x^2 \, z^2 + y^2 \, z^2 = 16 \, (n-1)^2 \, n^2 \, (n+1)^2 \, (n^8-4 \, n^6+22 \, n^4-4 \, n^2+1)^2

 

xy+xz+yz SQ 2

In two ways:

2881585^2 \; + \; 981552^2 \; + \; 164736^2 \; = \; 145^6
2964815^2 \; + \; 705024^2 \; + \; 83232^2 \; = \; 145^6

 
xy+xz+yz SQ 3

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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