For each positive integer
is defined to be the greatest integer such that, for every positive integer
can be written as the sum of
positive squares.
a.) Prove that
for each
.
b.) Find an integer
such that
.
c.) Prove that there are infintely many integers
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 $S(n)=n^{2}-14.$
Školjka