Solve the Diophantine equation x² + y² = z² in positive integers (Pythagorean triples).
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Step-by-Step Solution
Step 1: Introduce Primitive Pythagorean Triples
We are looking for positive integer solutions to the equation x2+y2=z2. These are known as Pythagorean triples. A common approach is to first find primitive Pythagorean triples, where x, y, and z are coprime. All other Pythagorean triples can be generated by multiplying a primitive triple by a common integer factor. The general form for primitive Pythagorean triples is given by Euclid's formula, where m and n are coprime positive integers of opposite parity, and m > n.
Step 2: Verify the Formula
Let's substitute the expressions for x, y, and z into the original equation to verify that they satisfy the Pythagorean theorem. This step confirms the validity of Euclid's formula.
Step 3: Expand and Simplify
Expanding the squares on the left side of the equation, we get (m4−2m2n2+n4) from (m2−n2)2 and 4m2n2 from (2mn)2. The right side is (m2+n2)2=m4+2m2n2+n4.
Step 4: Combine Terms
Combining the terms on the left side, −2m2n2+4m2n2 simplifies to 2m2n2. This makes the left side equal to the right side, confirming the identity.
Step 5: General Solution
The general solution for all Pythagorean triples (not just primitive ones) is obtained by multiplying the primitive triple formulas by an arbitrary positive integer k. Here, m and n are coprime positive integers, m > n, and m and n have opposite parity. k is any positive integer.