There is a cool identity relating Fibonacci numbers and Pythagorean triples, which states that given four consecutive Fibonacci numbers , the triple is a Pythagorean triple.
To prove this, let and . Then and . Substituting the abstract values to the equation
With a little simplification, we arrive at Euclid's formula for Pythagorean triples:
Check out my other notes at Proof, Disproof, and Derivation