Exercise 4 — Candidates who have followed the specialization course
We consider the equation (E) $$x ^ { 2 } - 5 y ^ { 2 } = 1$$ where $x$ and $y$ are natural integers.
Part A
We suppose that ( $x ; y$ ) is a solution pair of equation (E).
- Can $x$ and $y$ have the same parity? Justify.
- Prove that $x$ and $y$ are coprime.
- Let $k$ be a natural integer. Copy and complete the following table:
| \begin{tabular}{ l } Remainder of the euclidean |
| division of $k$ by 5 |
& 0 & 1 & 2 & 3 & 4 \hline | Remainder of the euclidean |
| division of $k ^ { 2 }$ by 5 |
& & & & & \hline \end{tabular}
- Deduce that $x \equiv 1$ [5] or $x \equiv 4$ [5].
Part B
Let $A$ be the matrix $\left( \begin{array} { c c } 9 & 20 \\ 4 & 9 \end{array} \right)$. We consider the sequences $\left( x _ { n } \right)$ and $\left( y _ { n } \right)$ defined by $$x _ { 0 } = 1 \text { and } y _ { 0 } = 0 \text {, and for all natural integer } n , \binom { x _ { n + 1 } } { y _ { n + 1 } } = A \binom { x _ { n } } { y _ { n } }$$
- For all natural integer $n$, express $x _ { n + 1 }$ and $y _ { n + 1 }$ in terms of $x _ { n }$ and $y _ { n }$.
- Prove by induction that, for all natural integer $n$, $\left( x _ { n } , y _ { n } \right)$ is a solution of equation (E).
- a. Determine $A ^ { 2 }$, then deduce $x _ { 2 }$ and $y _ { 2 }$. b. Let $p$ be a natural integer. Prove that if $y _ { p }$ is a multiple of 9 then $y _ { p + 2 }$ is also a multiple of 9. c. Deduce that $y _ { 2020 }$ is a multiple of 9.