1
$\begingroup$

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 $2$.

$\endgroup$
  • $\begingroup$ Yes, thanks! I have edited it $\endgroup$ – boltic92 May 14 at 0:24
  • $\begingroup$ Thanks, I've just ready it $\endgroup$ – boltic92 May 14 at 0:28
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.