Elementary Methods in Number Theory Exercise 1.4.10

Find all primes $p$ such that $29p+1$ is a square.

Proof:
\begin{equation}
29p+1=t^2
\end{equation}
So
\begin{equation}
29p=(t+1)(t-1)
\end{equation}
Both 29 and $p$ are primes,so
\begin{align*}
\begin{cases}
t+1=29\\
t-1=p\\
\end{cases}
\end{align*}(impossible)
or
\begin{align*}
\begin{cases}
t+1=p\\
t-1=29\\
\end{cases}
\end{align*}(p=31)
or
\begin{align*}
\begin{cases}
t+1=29p\\
t-1=1\\
\end{cases}
\end{align*}(impossible)
or
\begin{align*}
\begin{cases}
t-1=29p\\
t+1=1\\
\end{cases}
\end{align*}(impossible)
So $p=31$.