# Suranyi’s theorem

Suranyi’s theorem states that every integer $k$ can be expressed as the following sum:

 $k=\pm 1^{2}\pm 2^{2}\pm\cdots\pm m^{2}$

for some $m\in\mathbb{Z}^{+}$.
We prove this by induction, taking the first four whole numbers as our cases:

 $0=1^{2}+2^{2}-3^{2}+4^{2}-5^{2}-6^{2}+7^{2}$
 $1=1^{2}$
 $2=-1^{2}-2^{2}-3^{2}+4^{2}$
 $3=-1^{2}+2^{2}$

Now it suffices to prove that if the theorem is true for $k$ then it is also true for $k+4$.
As

 $(m+1)^{2}-(m+2)^{2}-(m+3)^{2}+(m+4)^{2}=4$

it’s simple to finish the proof:
if $k=\pm 1^{2}\pm\cdots\pm m^{2}$ then

 $(k+4)=\pm 1^{2}\pm\cdots\pm m^{2}+(m+1)^{2}-(m+2)^{2}-(m+3)^{2}+(m+4)^{2}$

and we are done.

Title Suranyi’s theorem SuranyisTheorem 2013-03-22 13:43:00 2013-03-22 13:43:00 mathcam (2727) mathcam (2727) 10 mathcam (2727) Theorem msc 11A99