For each positive integer
![\,n,\;S(n)\,](/media/m/4/8/d/48d97c7ee610141a96cd2b3e2d15f692.png)
is defined to be the greatest integer such that, for every positive integer
![\,k\leq S(n),\;n^{2}\,](/media/m/b/b/f/bbfe6215b5f5c4d313bdf9ca898d45e0.png)
can be written as the sum of
![\,k\,](/media/m/3/5/f/35f3b9ab836e4242a6f511c825a5958c.png)
positive squares.
a.) Prove that
![\,S(n)\leq n^{2}-14\,](/media/m/7/a/f/7afd5f6af6c5a039230212a3d8102308.png)
for each
![\,n\geq 4](/media/m/2/7/b/27bea78a22ec67bd9c753157e6099b9f.png)
.
b.) Find an integer
![\,n\,](/media/m/3/2/e/32e4ed37300f0cd666553950dba3bcad.png)
such that
![\,S(n)=n^{2}-14](/media/m/8/1/6/816beb245f406cce299d6f32ef652f2c.png)
.
c.) Prove that there are infintely many integers
![\,n\,](/media/m/3/2/e/32e4ed37300f0cd666553950dba3bcad.png)
such that
%V0
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
a.) Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
b.) Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
c.) Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$