
$\begin{aligned}[t]
7p+1 \text{ est un carré parfait} \iff \exists n\in\N,\ 7p+1=n^2\\
& \iff n^2-1=7p\\
& \iff (n+1)(n-1)=7p\\
& \iff \begin{cases}
n+1=7\\
n-1=p
\end{cases}\qquad\text{ou}\qquad\begin{cases}
n+1=p\\
n-1=7
\end{cases}\\
& \iff n=\frac{p+7}{2}.
\end{aligned}$
