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 $p=31$.