Does anyone have an idea about the following exercises? I have tried both of them by using induction, but I haven't got it.
Prove that for every $k\in{\mathbb{N}}$, if $4k=m_1^2+\dots m_{3+k}^2$ with $m_1,\dots,m_{3+k}$ positive integers, then (up to change the order) $m_1=m_2=m_3=m_4=1$ and the rest of $m_i$ are equal to $2$. And, if $4k+2=m_1^2+\dots m_{2+k}^2$, then $m_1=m_2=1$ and the rest of $m_i$ are equal to $2$.
They are not true. Note that $5\cdot 1+3\cdot 9=32=8\cdot 4$ so you can substitute five $1$s and three $3$s for eight $2$s. In particular, take $k=9$. You have $12$ numbers whose sum of squares is $36$. That can be four $1$s and eight $2$s (as the proposition says) or nine $1$s and three $3$s.