from typing import Tuple


def pyth_triple(u: int, v: int) -> Tuple[int, int, int]:
    """
    Generates a Pythagorean triple (u^2 - v^2, 2uv, u^2 + v^2).
    Conditions:
        1) u > v >= 1

    :param u: natural number
    :param v: natural number
    :return: A Pythagorean triple with a odd and b even.
    """
    if not u > v:
        raise ValueError("First argument must strictly be greater than the second.")
    elif not v >= 1:
        raise ValueError("Second argument must be greater than or equal to 1.")

    return u**2 - v**2, 2*u*v, u**2 + v**2