Pythagorean Triples Calculator
Check if three numbers form a Pythagorean triple or generate triples using Euclid's formula
Pythagorean Triples Calculator
Enter three positive integers to check if they form a Pythagorean triple (a² + b² = c²):
Example Calculations
Check: Famous 3-4-5 Triple
Numbers: 3, 4, 5
Calculation: 3² + 4² = 9 + 16 = 25
Verification: 5² = 25 ✅
Type: Primitive (GCD = 1)
Generate: Using m=2, n=1, k=1
Parameters: m=2, n=1, k=1
Formula: a=k(m²-n²), b=k(2mn), c=k(m²+n²)
Results: a=3, b=4, c=5
Type: Primitive triple
Common Pythagorean Triples
Primitive Triple Properties
GCD(a, b, c) = 1
Exactly one of a, b is odd
Exactly one of a, b is divisible by 3
Exactly one of a, b is divisible by 4
Exactly one of a, b, c is divisible by 5
Quick Tips
Three odd numbers cannot form a Pythagorean triple
Two even + one odd cannot form a triple
Valid triples: 3 even or 2 odd + 1 even
For primitive generation: GCD(m,n) = 1
Understanding Pythagorean Triples
What is a Pythagorean Triple?
A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the Pythagorean theorem: a² + b² = c². These represent the sides of a right triangle, where c is always the hypotenuse (longest side).
Primitive vs Non-Primitive
- •Primitive: GCD(a, b, c) = 1 (no common factors)
- •Non-primitive: Multiple of a primitive triple
- •Example: (3,4,5) is primitive, (6,8,10) is not
Euclid's Formula
a = k(m² - n²)
b = k(2mn)
c = k(m² + n²)
- m, n: Positive integers with m > n
- k: Positive scaling factor
- Primitive condition: GCD(m,n) = 1 and exactly one of m,n is odd, k = 1
Historical Note: This formula was known to ancient mathematicians and generates all primitive Pythagorean triples.