\section*{Exercise 4 — Candidates who have chosen the specialty course}
An integer $N$ is said to be a triangular number if there exists a natural number $n$ such that: $N = 1 + 2 + \ldots + n$.

For example, 10 is a triangular number because $10 = 1 + 2 + 3 + 4$.\\
The purpose of this problem is to determine triangular numbers that are perfect squares.\\
Recall that, for every non-zero natural number $n$, we have:

$$1 + 2 + \ldots + n = \frac{n(n+1)}{2}$$

\section*{Part A: triangular numbers and perfect squares}
\begin{enumerate}
  \item Show that 36 is a triangular number, and that it is also the square of an integer.
  \item a. Show that the number $1 + 2 + \ldots + n$ is the square of an integer if and only if there exists a natural number $p$ such that: $n^{2} + n - 2p^{2} = 0$.\\
b. Deduce that the number $1 + 2 + \ldots + n$ is the square of an integer if and only if there exists a natural number $p$ such that: $(2n + 1)^{2} - 8p^{2} = 1$.
\end{enumerate}

\section*{Part B: study of the associated Diophantine equation}
Consider (E) the Diophantine equation

$$x^{2} - 8y^{2} = 1$$

where $x$ and $y$ denote two integers.