help me!!! I will die!!!?

Standard contradiction:

Suppose some prime p divides both m²+n² and m³+n³.

Since "p" is prime, if it divides m³+n³ it must one of the two factors. {Can you factor m³+n³?}

Case 1) If "p" divides (m+n), it divides (m+n)² so it divides (m+n)²-(m²+n²) {Can you simplify?}, hence it divides one of the factors. But (m+n) is odd (m is even and n is not) so p is odd, so it does not divide "2". If "p" divides "m" then p divides (m+n)-m {Simplify?}. But (m,n)=1 means that no prime divides both "m" and "n", so this is not the case. What if "p" divides he other factor of (m+n)²-(m²+n²)? Can you finish?

Case 2) Very similar. I leave that to you.

If no prime divides both m²+n² and m³+n³, what is (m²+n², m³+n³) ?