\[n, \; m\]
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 \[2=1^1+1^2\]
. \[\frac{58}{2}=29=5^2+2^2\]
.