📐 Pythagorean Triples Calculator
Generate, check, or find Pythagorean triples
How to Use This Calculator
Choose Operation
Select whether you want to generate a triple (using m and n), check if three numbers form a triple, or find triples containing a specific side.
Enter Values
Input the required values based on the operation you selected. For generation, enter m and n (m > n). For checking, enter all three sides.
Calculate
Click the button to generate, check, or find Pythagorean triples.
Review Results
See the generated triple, verification result, or list of triples containing your specified side.
Formula
Euclid's Formula: a = m² - n², b = 2mn, c = m² + n²
where m > n > 0, and m and n are coprime (gcd = 1) for primitive triples
Pythagorean Triple:
Three positive integers (a, b, c) such that a² + b² = c²
Primitive Triple:
A triple where a, b, and c have no common factors (gcd(a, b) = 1)
Common Examples:
- (3, 4, 5) - Primitive
- (5, 12, 13) - Primitive
- (8, 15, 17) - Primitive
- (6, 8, 10) - Not primitive (multiple of 3-4-5)
- (9, 12, 15) - Not primitive (multiple of 3-4-5)
Checking if Triple:
Verify: a² + b² = c² (where c is the largest number)
About Pythagorean Triples Calculator
The Pythagorean Triples Calculator generates, checks, and finds Pythagorean triples. A Pythagorean triple consists of three positive integers (a, b, c) that satisfy the equation a² + b² = c², forming the sides of a right triangle.
When to Use This Calculator
- Mathematics: Generate and study Pythagorean triples
- Education: Learn about number theory and right triangles
- Construction: Find integer side lengths for right triangles
- Problem Solving: Find triples for mathematical puzzles
- Verification: Check if three numbers form a Pythagorean triple
Why Use Our Calculator?
- ✅ Multiple Operations: Generate, check, or find triples
- ✅ Euclid's Formula: Uses the standard method for generation
- ✅ Primitive Detection: Identifies primitive vs. non-primitive triples
- ✅ Verification: Checks if numbers satisfy the Pythagorean theorem
- ✅ Free: No registration required
Key Concepts
- Primitive Triple: When gcd(a, b) = 1, the triple is primitive
- Scaled Triples: Multiply a primitive triple by any integer k to get more triples
- Euclid's Formula: Generates all primitive triples when m and n are coprime
- Even and Odd: In a primitive triple, exactly one of a and b is even
Frequently Asked Questions
What is a Pythagorean triple?
A Pythagorean triple is three positive integers (a, b, c) such that a² + b² = c². These represent the side lengths of a right triangle.
What's the difference between primitive and non-primitive triples?
A primitive triple has no common factors (gcd(a, b, c) = 1). Non-primitive triples are multiples of primitive ones. Example: (3, 4, 5) is primitive, (6, 8, 10) is not.
How does Euclid's formula work?
Given integers m > n > 0: a = m² - n², b = 2mn, c = m² + n². If m and n are coprime and one is even, this generates a primitive triple.
Are all Pythagorean triples generated by Euclid's formula?
Euclid's formula generates all primitive triples. All other triples are multiples (k × primitive triple) of these.
Can I get a triple if m and n are not coprime?
Yes, but it won't be primitive. The resulting triple will have a common factor that can be divided out to get the primitive version.