If Two Numbers are Expressible as Sums of Two Squares and One Divides the Other, Is the Result Also Expressible As a Sum of Two Squares?

If  
\[n, \; m\]
  can each be written as the sum of two squares and  
\[m\]
  divides  
\[n\]
, can we write  
\[\frac{n}{m}\]
  as the sum of two squares?
Yes it is. Each prime  
\[p\]
  of the form  
\[4k+3\]
  which divides  
\[n, \;m\]
  must occur as even powers, with the power in the decomposition of  
\[n\]
  greater than in the decomposition of  
\[m\]
.
The result follows using When an Integer Can be Expressed as a Sum of Two Squares.
For example,  
\[58=7^2+3^2\]
  and is divisible by  
\[2=1^1+1^2\]
.  
\[\frac{58}{2}=29=5^2+2^2\]
.

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