## 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?

\[n, \; m\]

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 \[2=1^1+1^2\]

. \[\frac{58}{2}=29=5^2+2^2\]

.