Disprove the following statements:
(a) If $m$ is a sum of two squares and $m \mid n$ then $n$ is also a sum of two squares.
(b) If both $m$ and $n$ can be expressed as sum of two squares then $m + n$ can also be written as sum of two squares.
(c) Assume all three positive distinct integers $n_1$, $n_2$, and $n_3$ cannot be expressed as a sum of two squares. Therefore, $n_1 + n_2 + n_3$ cannot be expressed as a sum of two squares.
We will disprove the statements with counter-examples.
(a) Let $m=5=2^2 + 1^2$, and $n=15$.
We have $m \mid n$. But $15=3 \times 5$ and $3 \equiv 3 \pmod 4$, and so by Corollary (8.8) we can't write $n$ as the sum of two squares.
(b) Let $m=2=1^1+1^2$ and $n=5=2^2+1^2$.
Here $m+n=7$, a prime congruent to 3 modulo 4, and so by Corollary (8.8) we can't write $m+n$ as the sum of two squares.
(c) Let $n_1=3, n_2=7, n_3=15$.
Here $n_1-3$ and $n_2=7$ are primes congruent to 3 modulo 4 and so can't be written as a sum of two squares by Theorem (8.5).
Similarly $n_3=15=3 \times 5$ has a single prime factor 3 congruent to 3 modulo 4, and so can't be written as a sum of two squares by Theorem (8.5).
The sum $n_1+n_2+n_3=25=5^2 + 0^2$, and so can be written as the sum of two squares.