Prove that there are infinitely many natural numbers that cannot be written as the sum of $6$ natural number squares.

I am stumped on this problem. The furthest I have got is to assume that there exists a largest number $$N$$ that cannot be written as the sum of $$6$$ squares. Ideally, I was trying to get a contradiction by constructing a larger number, dependant on $$N$$, that cannot be written as the sum of $$6$$ squares, but I have gotten nowhere with that.

even if you don't allow 0 it's not true. Legendre's theorem says you can write a number as the sum of three squares iff it isn't in the form $$4^n(8k+7)$$.for n write it as the sum of $$m+t$$ such that m is sum of three nonzero square and t is of the form $$8k+3$$ which must be sum of three nonzero square. the only problem occurs when $$n=8k+10=12+(8(k-1)+6)$$ and again each of $$12$$ and $$8k+6$$ are sum of three nonzero square.