How Pythagorean triples work
A Pythagorean triple is a set of positive integers (a, b, c) satisfying a² + b² = c², where c is the hypotenuse. To check a candidate, order its sides and square them: 7² + 24² = 49 + 576 = 625 = 25².
Primitive and scaled triples
A triple is primitive when gcd(a,b,c)=1. Thus (3,4,5) is primitive, while (12,16,20) has GCD 4 and is the scaled triple 4×(3,4,5).
Euclid’s formula and its exact conditions
For integers m>n>0, the scaled formula is a=k(m²−n²), b=2kmn, c=k(m²+n²). It produces a primitive triple precisely when gcd(m,n)=1, m and n have opposite parity, and k=1. Other valid inputs can produce non-primitive triples.
Algebraic check: (m²−n²)²+(2mn)² = m⁴−2m²n²+n⁴+4m²n² = m⁴+2m²n²+n⁴ = (m²+n²)².
Finding a triple containing a known side
Select “Find a known leg” or another finder option. A side can belong to more than one triple, so the tool searches every Euclid seed and scale up to a sufficient calculated bound and reports that bound. For example, leg 20 has four matches through (20,99,101), not just the matches below the old default c≤200 by coincidence.