A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive).  How many lattice points lie on the graph of the equation $x^2-y^2=47$?
I notice that this equation can be factored as $(x+y)(x-y)=47$. This means that $x+y$ and $x-y$ must be factors of 47, which is a prime number. The only factors of 47 are $\pm 1$ and $\pm 47$. So, I have four possible cases: $x+y=1$ and $x-y=47$, $x+y=47$ and $x-y=1$, $x+y=-1$ and $x-y=-47$, and $x+y=-47$ and $x-y=-1$. I can solve for $x$ and $y$ in each case by adding or subtracting the equations. For example, in the first case, if I add the equations, I get $2x=48$, so $x=24$. Then, subtracting the equations, I get $2y=-46$, so $y=-23$. Similarly, in the second case, I get $x=24$ and $y=23$, in the third case, I get $x=-24$ and $y=23$, and in the fourth case, I get $x=-24$ and $y=-23$. So, I have found four pairs of integers that satisfy the equation: $(24,-23)$, $(24,23)$, $(-24,23)$, and $(-24,-23)$. Therefore, there are four lattice points on the graph of the equation. 