The Pythagorean Theorem, arguably one of the best-known results in mathematics, states that a triangle is a right triangle if and only if the sum of the squares of the lengths of two of its sides equals the square of the length of its third side. Closely associated with the Pythagorean Theorem is the concept of Pythagorean triples. A "Pythagorean triple" is a triple of numbers (a, b, c) such that a[squared] + b[squared] = c[squared], where a, b, and c, are positive integers. A simple way to prove this "3, 4, 5 connection (with respect to primitive Pythagorean triples)" involves a nice application of the notions of congruence and modular arithmetic from elementary number theory. This article provides the proof of primitive Pythagorean triples that is accessible to students who are familiar with, or are mathematically mature enough to understand its definition. This article concludes with a list of the 16 primitive Pythagorean triples.