I had the pleasure of filling in for a colleague in her Discrete Structure class, which is basically our introduction to proof class. Currently they are covering some introductory concepts from Number Theory. One of my favorite results had been covered in the class just before I needed to fill in: The Euclidean Algorithm. Anyways, toward the end of class we introduced the concept of Pythagorean Triples.
Def: An ordered triple, [tex](a, b, c)[/tex] of positive integers such that [tex]a^2 + b^2 = c^2[/tex] is called a pythagorean triple.
EX: (3, 4, 5)
EX: (6, 8, 10)
EX: (9, 12, 15)
EX: (5, 12, 13)
Notice that the first three examples are of the same “type.” That is, the second and the third are multiples of the first. In fact, we could state the following theorem.
Thm: If [tex](a, b, c)[/tex] is a Pythagorean triple then for any positive integer [tex]k[/tex], [tex](ka, kb, kc)[/tex] is also a Pythagorean triple.
Now, the last of my examples is fundamentally different from the other three. In fact, we could create a new definition.
Def: If the greatest common divisor of the pythagorean triple [tex](a, b, c)[/tex] is [tex]1[/tex], then it is called a primitive pythagorean triple. (In other words, if they have no common factors it is a primitive pythagorean triple)
I left the class with an question and recommended that if they were interested they should do a little research to find the answer.
Question: How many primitive pythagorean triples are there? That is, are there infinitely many or only some finite number of them?
Here are some interesting facts from Wikipedia about Pythagorean triples:
- 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), (a?b) is divisible by 7
- At most one of a, b is a square
- The hypotenuse, c, is an odd number
- Every integer greater than 2 is part of a Pythagorean triple
- The area (A=ab/2) is not an integer
- For any Pythagorean triple, the product of the two nonhypotenuse legs is always divisible by 12, and the product of all three sides is divisible by 60.
- There exist infinitely many primitive Pythagorean triples whose hypotenuses are squares of natural numbers.
- There exist infinitely many primitive Pythagorean triples of which one of the arms is the square of a natural number.
- For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
- For each natural number n, there exist at least n different Pythagorean triples with the same arm a, where a is some natural number
- For each natural number n, there exist at least n different triangles with the same hypotenuse.
- In every Pythagorean triple, the radius of the in-circle and the radii of the three ex-circles are natural numbers.
- There is no Pythagorean triple of which the hypotenuse and one arm are the arms of another Pythagorean triple.