Pythagorean Triples Explorer
Discover patterns in primitive Pythagorean triples through observation, expression, and proof
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Fundamentals of Mathematics
Primitive Pythagorean Triples
Three positive integers (a, b, c) where:
• a² + b² = c²
• gcd(a, b, c) = 1 (no common factor)
Example: (3, 4, 5)
Generation Formula
All primitive triples have the form:
(m²−n², 2mn, m²+n²)
where m > n > 0, gcd(m,n) = 1,
and exactly one of m, n is even
Match Non-Primitive to Primitive
Every non-primitive triple is a multiple of a primitive triple. Can you find the matches?
Non-Primitive
Primitive
Explore Specific Properties
Test Your Own Property
Examples
Select a property above to see examples
Express Your Mathematical Observations
Based on your exploration, write a mathematical statement about primitive Pythagorean triples:
- Use precise quantifiers: "for all," "for every," "there exists," "exactly one"
- Be specific: "primitive Pythagorean triple" not just "Pythagorean triple"
- State clear properties: divisibility, parity, magnitude relationships
- Avoid vague terms: instead of "many," use "all" or give specific conditions
Select a statement from the "Say It" tab first
Arrange the Proof Steps
Click "Generate Proof Steps" to begin