Trigonometry/For Enthusiasts/Pythagorean Triples

Pythagorean TriplesEdit

A Pythagorean triple has three positive integers a, b, and c, such that a2 + b2 = c2. In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.[1] Evidence from megalithic monuments on the Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c).


Generating Pythagorean TriplesEdit

The integers

 a = m^2 - n^2 ,\ \, b = 2mn ,\ \, c = m^2 + n^2

always form a Pythagorean triple, that is

\displaystyle a^2 + b^2 = c^2


Show it works
  • (easy) Show that the formula is true whatever integer value we put for m and n.
How was it discovered?
  • (hard) How would someone find such a formula for generating Pythagorean Triples in the first place?
    • Don't worry if you don't come up with an answer to this. Just investigating the question will help you practice with algebra.


Examples of Pythagorean TriplesEdit

Some well-known examples are (3, 4, 5) and (5, 12, 13).

A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).

The following is a list of primitive Pythagorean triples with values less than 100:

(3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53), (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73), (65, 72, 97)
Are they all generated?
  • Does the formula for generating Pythagorean Triples generate all the triples shown?
Fermat's last theorem
  • What is Fermat's Last Theorem?
    • What goes wrong if you try to use and adapt the formula for Pythagorean Triples for it?

ReferencesEdit

  1. Needs a reference
Last modified on 4 January 2011, at 20:26