# Equalities with sum of squares

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 $$2$$.

• Yes, thanks! I have edited it – boltic92 May 14 at 0:24
• Thanks, I've just ready it – boltic92 May 14 at 0:28

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.