Say a^2 + b^2 and a + b were each divisible by some prime number P. So
a^2 + b^2 = PX
a + b = PY
for some integers, X,Y
Since one of a, b is even, one odd, P is not 2.
Then 2ab = (PY^2)  PX = P(PY  X) is also divisible by P. So P must be one of the prime factors occurring either in a or b. Assume, without losing generality, that a = PZ then
b = PY  a = P(Y  Z), so that both a and b share the same common factor, namely P. A contradiction.

Posted by FrankM
on 20210805 11:19:23 