Pythagorean Triples

[lpetrich]

 Pythagorean theorem - Pythagoras's Theorem - a theorem with numerous proofs, a theorem that has been at least partially recognized for many centuries in the literate parts of the world. Though named after  Pythagoras - Pythagoras (Stanford Encyclopedia of Philosophy), it is doubtful whether he himself came up with a proof of it, and it must be noted that it is difficult to distinguish fact from fiction about him. Was he a vegetarian? Did he eat the meat of sacrificed animals? Did he sacrifice animals? However, the surviving sources on him agree that he forbade the eating of beans. His followers formed a sort of religious cult or philosophical school,  Pythagoreanism - Pythagoreanism (Stanford Encyclopedia of Philosophy), and Plato was greatly influenced by that.

For side lengths a, b, c of a right triangle, where a and b are the lengths of the sides next to the right angle and c the length of the side opposite that angle, the hypotenuse, it is

a² + b² = c²

For a spherical surface with radius R,
cos(a/R)*cos(b/R) = cos(c/R)

For a hyperbolic surface with radius factor R,
cosh(a/R)*cosh(b/R) = cosh(c/R)

 

[lpetrich]

A  Pythagorean triple is a triplet of integers a, b, c that satisfies Pythagoras's relation
a² + b² = c²

If a, b, c are a Pythagorean triple, then for integer k, the triplet k*a, k*b, k*c are also a Pythagorean triple. In fact, if a, b, c are relatively prime or coprime, then they form a "primitive" Pythagorean triple.

Some of them have been known for some millennia, like 3,4,5 and 5,12,13.

 Formulas for generating Pythagorean triples includes one from Euclid's Elements that generates all primitive triples for integer n and m:

a = n² - m²
b = 2*n*m
c = n² + m²

For a primitive triple, n and m must be coprime and one of them must be even.

This formula does not generate all triples, however, an example being 9,12,15. That one is 3*(3,4,5).

 

[lpetrich]

There are several other formulas.

Here is the Fibonacci-Stiefel formula, as it may be called:
2n+1, 2n(n+1), 2n(n+1)+1
Equivalent to Euclid's formula for n, n+1

Another one is the Dickson formula:
r+s, r+t, r+s+t
for integers r,s,t the satisfy r² = 2*s*t

Etc.