Exercise 5 -- For candidates who have followed the specialization course
A right-angled triangle with integer sides (TRPI) is a right-angled triangle whose three sides have lengths that are natural integers. If the triangle with sides $x$, $x+1$ and $y$, where $y$ is the length of the hypotenuse, is a TRPI, we will say that the couple $(x ; y)$ defines a TRPI.
Part A
Prove that the couple of natural integers $(x ; y)$ defines a TRPI if, and only if, we have: $$y^{2} = 2x^{2} + 2x + 1$$
Show that the TRPI having the smallest non-zero sides is defined by the couple $(3 ; 5)$.
a. Let $n$ be a natural integer. Show that if $n^{2}$ is odd then $n$ is odd. b. Show that in a couple of integers $(x ; y)$ defining a TRPI, the number $y$ is necessarily odd.
Show that if the couple of natural integers $(x ; y)$ defines a TRPI, then $x$ and $y$ are coprime.
Part B
We denote by $A$ the square matrix: $A = \left( \begin{array}{ll} 3 & 2 \\ 4 & 3 \end{array} \right)$, and $B$ the column matrix: $B = \binom{1}{2}$. Let $x$ and $y$ be two natural integers; we define the natural integers $x^{\prime}$ and $y^{\prime}$ by the relation: $$\binom{x^{\prime}}{y^{\prime}} = A\binom{x}{y} + B.$$
Express $x^{\prime}$ and $y^{\prime}$ as functions of $x$ and $y$. a. Show that: $y^{\prime 2} - 2x^{\prime}(x^{\prime}+1) = y^{2} - 2x(x+1)$. b. Deduce that if the couple $(x ; y)$ defines a TRPI, then the couple $(x^{\prime} ; y^{\prime})$ also defines a TRPI.
We consider the sequences $(x_{n})_{n \in \mathbb{N}}$ and $(y_{n})_{n \in \mathbb{N}}$ of natural integers, defined by $x_{0} = 3$, $y_{0} = 5$ and for every natural integer $n$: $$\binom{x_{n+1}}{y_{n+1}} = A\binom{x_{n}}{y_{n}} + B.$$ Show by induction that, for every natural integer $n$, the couple $(x_{n} ; y_{n})$ defines a TRPI.
Determine, by the method of your choice which you will specify, a TRPI whose side lengths are greater than 2017.
\section*{Exercise 5 -- For candidates who have followed the specialization course}
A right-angled triangle with integer sides (TRPI) is a right-angled triangle whose three sides have lengths that are natural integers. If the triangle with sides $x$, $x+1$ and $y$, where $y$ is the length of the hypotenuse, is a TRPI, we will say that the couple $(x ; y)$ defines a TRPI.
\section*{Part A}
\begin{enumerate}
\item Prove that the couple of natural integers $(x ; y)$ defines a TRPI if, and only if, we have:
$$y^{2} = 2x^{2} + 2x + 1$$
\item Show that the TRPI having the smallest non-zero sides is defined by the couple $(3 ; 5)$.
\item a. Let $n$ be a natural integer. Show that if $n^{2}$ is odd then $n$ is odd.\\
b. Show that in a couple of integers $(x ; y)$ defining a TRPI, the number $y$ is necessarily odd.
\item Show that if the couple of natural integers $(x ; y)$ defines a TRPI, then $x$ and $y$ are coprime.
\end{enumerate}
\section*{Part B}
We denote by $A$ the square matrix: $A = \left( \begin{array}{ll} 3 & 2 \\ 4 & 3 \end{array} \right)$, and $B$ the column matrix: $B = \binom{1}{2}$. Let $x$ and $y$ be two natural integers; we define the natural integers $x^{\prime}$ and $y^{\prime}$ by the relation:
$$\binom{x^{\prime}}{y^{\prime}} = A\binom{x}{y} + B.$$
\begin{enumerate}
\item Express $x^{\prime}$ and $y^{\prime}$ as functions of $x$ and $y$.\\
a. Show that: $y^{\prime 2} - 2x^{\prime}(x^{\prime}+1) = y^{2} - 2x(x+1)$.\\
b. Deduce that if the couple $(x ; y)$ defines a TRPI, then the couple $(x^{\prime} ; y^{\prime})$ also defines a TRPI.
\item We consider the sequences $(x_{n})_{n \in \mathbb{N}}$ and $(y_{n})_{n \in \mathbb{N}}$ of natural integers, defined by $x_{0} = 3$, $y_{0} = 5$ and for every natural integer $n$:
$$\binom{x_{n+1}}{y_{n+1}} = A\binom{x_{n}}{y_{n}} + B.$$
Show by induction that, for every natural integer $n$, the couple $(x_{n} ; y_{n})$ defines a TRPI.
\item Determine, by the method of your choice which you will specify, a TRPI whose side lengths are greater than 2017.
\end{enumerate}