1. Pythagorean Triples Big Idea: Pythagorean triple is any set of three whole numbers that satisfies the Pythagorean theorem c2 = a2 + b2. A primitive Pythagorean triple has the additional characteristic that the greatest common divisor of a, b, and c is 1. Goal: Understand and recognize primitive Pythagorean triples and their elementary properties; use Euclids’s formula to generate primitive Pythagorean triples; prove Euclid’s formula.
2. Pythagorean Triples Any set of three whole numbers that satisfies the Pythagorean theorem is called a Pythagorean triple. Examples of Pythagorean triples include {3, 4, 5}, {6, 8, 10}, and {5, 12, 13}. (Pythagorean Triples are generally denoted in brackets of the form {a,b,c}. Notice that {3, 4, 5} and {6, 8, 10} represent similar triangles. The sides of similar right triangles are proportional to each other. So multiples of Pythagorean triples are also Pythagorean triples. For any given Pythagorean triple there exists an infinite number of triples of the form k{a,b,c}, where k is a constant. 10 5 3 6 4 8
3. Pythagorean Triples Whole number triples whose greatest common divisor is 1 are considered primitive. Thus {3, 4, 5} is a primitive Pythagorean triple and {6, 8, 10} is not. Integers a and b whose greatest common factor is 1 are said to be coprime or relatively prime. For example: 6 and 35 are coprime, but 6 and 27 are not because they are both divisible by 3.
4. Pythagorean Triples Let’s try to find additional primitive Pythagorean triples. In other words, find a set of positive integers a, b, and c such that a 2 + b 2 = c 2where a, b, and c are coprime. “Guess & check” is one strategy to find Pythagorean triples, but it is time consuming. Euclid developed a formula for finding such primitives. The formula states that an arbitrary pair of positive integers m and n with m> n will form a Pythagorean triple if m and n are coprime and one of them is odd. The formula is: a = m2 – n2 b = 2mn c = m2 + n2.
5. Pythagorean Triples Activity PT #1 Use Euclid’s formula to find three additional primitive Pythagorean triples other than {3,4,5} and {5,12,13}. Activity PT #2 Use basic algebra to prove Euclid’s formula that c2 = a2 + b2 when a = m2 – n2, b = 2mn and c = m2 + n2.
6. Pythagorean Triples Elementary properties of primitive Pythagorean triples (c-a)(c-b)/2 is always a perfect square. Exactly one of a, b is odd; c is odd. Exactly one of a, b is divisible by 3. Exactly one of a, b is divisible by 4. Exactly one of a, b, c is divisible by 5. Exactly one of a, b, (a + b), (b – a) is divisible by 7. Exactly one of (a + c), (b + c), (c – a), (c – b) is divisible by 8. Exactly one of (a + c), (b + c), (c – a), (c – b) is divisible by 9. Exactly one of a, b, (2a + b), (2a – b), (2b + a), (2b – a) is divisible by 11. The hypotenuse exceeds the even leg by the square of an odd integer j, and exceeds the odd leg by twice the square of an integer k>0, from which it follows that: There are no primitive Pythagorean triples in which the hypotenuse and a leg differ by a prime number greater than 2.
7. Pythagorean Triples Activity PT #3 Select five of the elementary properties of primitive Pythagorean triples and prove that they hold true for the three primitive Pythagorean triples you created in Activity #1.
8. Pythagorean Triples A bit of history… Around 4000 years ago, the Babylonians and the Chinese used the concept of the Pythagorean triple {3, 4, 5} to construct a right triangle by dividing a long string into twelve equal parts, such that one side of the triangle is three, the second side is four and the third side is five sections long. In India (8th - 2nd century BC), the BaudhayanaSulba Sutra contained a list of Pythagorean triples, a statement of the theorem and the geometrical proof of the theorem for an isosceles right triangle. Pythagoras (569–475 BC), used algebraic methods to construct Pythagorean triples. He was not universally credited with this for another 500 years. The ancient Greek philosopher Plato (c. 380 BC) used the expressions 2n, n2 – 1, and n2 + 1 to produce Pythagorean triples.
