Example 1. If in a Pythagorean triple, the difference between the hypotenuse and one of the legs is 1, then the form of the Pythagorean triple is
$$
2 a+1,2 a^{2}+2 a, 2 a^{2}+2 a+1 .
$$
Proof: Let the hypotenuse be $c$, the other leg be $c-1$, and the base be $x$.
Since $(c, c-1)=1$, $x, c-1, c$ form a set of primitive Pythagorean triples. Also, since $c$ is odd, $c-1$ is even, thus $x$ is odd.
Let $x=2a+1$, then we have
$(2a+1)^{2}+(c-1)^{2}=c^{2}$.
Solving for $c$ gives $c=2a^{2}+2a+1$,
$$
c-1=2a^{2}+2a \text{. }
$$

Therefore, the Pythagorean triple has the form
$$
2a+1, 2a^{2}+2a, 2a^{2}+2a+1.
$$