(i) Suppose $x , y$, and $z$ are whole numbers such that $x ^ { 2 } - 19 y ^ { 2 } = z$. Show that for any such $x , y$ and $z$, it is true that $$\left( x ^ { 2 } + N y ^ { 2 } \right) ^ { 2 } - 19 ( 2 x y ) ^ { 2 } = z ^ { 2 }$$ where $N$ is a particular whole number which you should determine. (ii) Find $z$ if $x = 13$ and $y = 3$. Hence find a pair of whole numbers ( $x , y$ ) with $x ^ { 2 } - 19 y ^ { 2 } = 4$ and with $x > 2$. (iii) Hence find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 19 y ^ { 2 } = 1$ and with $x > 1$. Is your solution the only such pair of positive whole numbers $( x , y )$ ? Justify your answer. (iv) Prove that there are no whole number solutions $( x , y )$ to $x ^ { 2 } - 25 y ^ { 2 } = 1$ with $x > 1$. (v) Find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 17 y ^ { 2 } = 1$ and with $x > 1$. This page has been intentionally left blank
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\section*{2. For ALL APPLICANTS.}
(i) Suppose $x , y$, and $z$ are whole numbers such that $x ^ { 2 } - 19 y ^ { 2 } = z$. Show that for any such $x , y$ and $z$, it is true that
$$\left( x ^ { 2 } + N y ^ { 2 } \right) ^ { 2 } - 19 ( 2 x y ) ^ { 2 } = z ^ { 2 }$$
where $N$ is a particular whole number which you should determine.\\
(ii) Find $z$ if $x = 13$ and $y = 3$. Hence find a pair of whole numbers ( $x , y$ ) with $x ^ { 2 } - 19 y ^ { 2 } = 4$ and with $x > 2$.\\
(iii) Hence find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 19 y ^ { 2 } = 1$ and with $x > 1$.
Is your solution the only such pair of positive whole numbers $( x , y )$ ? Justify your answer.\\
(iv) Prove that there are no whole number solutions $( x , y )$ to $x ^ { 2 } - 25 y ^ { 2 } = 1$ with $x > 1$.\\
(v) Find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 17 y ^ { 2 } = 1$ and with $x > 1$.
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