Find prime numbers : $p_1,p_2,\cdots,p_8$ satisfying : $p_1^2+\cdots+p_7^2=p_8^2$.
Form test in class of my brother .
We already have an example with repeats. Assume there are no repeats; now look at the equation modulo 4. Quadratic residues: $0,1$.
If 2 does not appear on the left hand side, then everything is congruent to 1 there and so we obtain $7\equiv 3$ in total. This is impossible.
If 2 does, then we have $4+6\equiv 2$. This is also impossible.
Hence there are no solutions where the LHS comes out as being a perfect square of anything in this case!
Take $p_1=p_2=\cdots=p_6=2$. You don't have to search very far to find $p_7=5$, which gives LHS=$49=7^2$.