Pythagorean triple: (a,b,c) with a²+b²=c². Primitive: gcd(a,b,c)=1. Euclid's formula: a=m²−n², b=2mn, c=m²+n² for m>n>0, coprime, opposite parity. First: (3,4,5), (5,12,13), (8,15,17), (7,24,25)... Infinitely many exist.
How to Use
Enter m and n (or check a triple):
- Generate: Via Euclid's formula
- Check: Verify a²+b²=c²
- Primitive?: gcd=1 check
Euclid's Formula
For coprime m>n>0 with m−n odd: a=m²−n², b=2mn, c=m²+n² gives ALL primitive triples. Non-primitive: multiply by k. This parametrizes the entire set of Pythagorean triples.
Triple Tree
Berggren (1934): all primitive triples form a ternary tree rooted at (3,4,5). Three matrix transformations generate all children. Every primitive triple appears exactly once.
Step-by-Step Instructions
- 1Enter m and n.
- 2Generate triple.
- 3Check primitive.
- 4List multiples.
- 5Explore tree.