Complete proof of Pythagorean triple Conjecture

For the Pythagorean triple conjecture, we adopted proof by contradiction, transformation method of propositions, and based on the congruence descending method arithmetic properties. On this basis, we have deduced necessary condition under which the conjecture does not hold, namely congruence expressions. As long as we can prove that these congruence expressions are groundless, we can solve this conjectue completely that conjecture. In addition, we delivered the complete proof indicating that the Pythagorean triple holds.