a^2+b^2, a^2+c^2, b^2+c^2

 
 
a^2 \; + \; b^2 \; = \; x^2
a^2 \; + \; c^2 \; = \; y^2
b^2 \; + \; c^2 \; = \; z^2

 
Here are the parametric equations:

a \; = \; | \:(4 \, p^2 - r^2) \, q \:|
b \; = \; 4 \, p \, q \, r
c \; = \; | \:(4 \, q^2 - r^2) \, p \:|

where   (p, \; q, \; r)   is a Pythagorean triple

p \; = \; m^2 \; - \; n^2
q \; = \; 2 \, m \, n
r \; = \; m^2 \; + \; n^2

a = (4 \, (m^2 - n^2)^2 - (m^2 + n^2)^2) \, (2 \, m \, n) = 6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5

b = 4 \,(m^2 - n^2)(2 \, m \, n) \,(m^2 + n^2) = 8 \, m^5 \, n - 8 \, m \, n^5

c = (4 \, (2 \, m \, n)^2 - (m^2 + n^2)^2) \, (m^2 - n^2) = -m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6

a^2 + b^2
= (6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5)^2 + (8 \, m^5 \, n - 8 \, m \, n^5)^2
= 4 \, (5 \, m^5 \, n - 6 \, m^3 \, n^3 + 5 \, m \, n^5)^2

a^2 + c^2
= (6 \, m^5 \, n - 20 \, m^3 \, n^3 + 6 \, m \, n^5)^2 + (-m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6)^2
=  (m^2 + n^2)^6

b^2 + c^2
= (8 \, m^5 \, n - 8 \, m \, n^5)^2 + (-m^6 + 15 \, m^4 \, n^2 - 15 \, m^2 \, n^4 + n^6)^2
= (m^6 + 17 \, m^4 \, n^2 - 17 \, m^2 \, n^4 - n^6)^2

 

PPT SQUARES 1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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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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