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Integer as sum of 6 squares.

Can every integer be written as sum of exactly 6 squares? I am also curious to know when an integer can be written as sum of exactly 8 squares. I know the problem is related to Waring-Hilbert ...
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Generalized Pell's equation

We know that Let $d$ be a positive square free integer and $r$ an integer satify $r^2+|r|\le d$. Suppose $x$ and $y$ are positive integers that satify $x^2-dy^2=r$. Then $\frac xy$ is a convergent to ...
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To find all integral solutions of $3x^2 - 4y^2 = 11$

I have to find all integral solutions of $3x^2 - 4y^2 = 11$. I looked at the equation $\text{mod}\ 3$, and $\text{mod}\ 4$ to see that $x^2 \equiv 1\ \text{mod}\ 4$ and $y^2 \equiv 1\ \text{mod}\ 3$ ...
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Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 adjacent ...
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Positive integer $n$ such that $2n+1$ , $3n+1$ are both perfect squares
How many positive integer $n$ are there such that $2n+1$ , $3n+1$ are both perfect squares ? $n=40$ is a solution . Is this the only solution ? Is it possible to tell whether finitely many or ...