What is the minimum n to get a prime pythagorean n-tuple?

Question

I want the minimum n for the equation:

$$X_1^2 + X_2^2 + . . . X_{n-1}^2 = X_n^2$$

Such that the solution only has prime numbers. It easy to check that n > 3. But, what is the minimum n?

Answer 1

There is no solution for $n=4$ (hint: consider the equation mod $4$), but for $n=5$ there is $$ 5^2 = 2^2 + 2^2 + 2^2 + 2^2 + 3^2 $$