Trigonometry/For Enthusiasts/Pythagorean Triples

Pythagorean Triples

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. 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 Triples

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 Triples

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: 9:(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?