(a*p + b*q + c*r), (a*q – b*p), (a*r – c*p), (b*r – c*q) — Part 2

 
 
Can you find positive integers   a, b, c, p, q, r   so that the expressions

a \, p \; + \; b \, q \; + \; c \, r
a \, q \; - \; b \, p
a \, r \; - \; c \, p
b \, r \; - \; c \, q

are to be made squares.

 
 
Note that

(a \, p + b \, q + c \, r)^2 \; + \; (a \, q - b \, p)^2 \; + \; (a \, r - c \, p)^2 \; + \; (b \, r - c \, q)^2

= \; a^2 \, p^2 + a^2 \, q^2 + a^2 \, r^2 + b^2 \, p^2 + b^2 \, q^2 + b^2 \, r^2 + c^2 \, p^2 + c^2 \, q^2 + c^2 \, r^2

= \; (a^2 + b^2 + c^2) \,(p^2 + q^2 + r^2)

 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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