(a^2 + b^2 + c^2 + d^2)(p^2 + q^2 + r^2 + s^2) — Part 3

 
 
(a^2 + b^2 + c^2 + d^2) \,(p^2 + q^2 + r^2 + s^2)

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

= \; (ap - bq - cr - ds)^2 + (aq + bp + cs - dr)^2 + (ar + cp + dq - bs)^2 + (as + dp + br - cq)^2

[ a sum of four squares]

Or

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

[ as a sum of seven squares ]

 
 

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

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

are to be made squares.
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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