## 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
$2=1^1+1^2$
.
$\frac{58}{2}=29=5^2+2^2$
.