1. The problem is to find a Pythagorean triple, which is a set of three positive integers $a$, $b$, and $c$ such that they satisfy the Pythagorean theorem: $$a^2 + b^2 = c^2$$ 2. The formula to generate Pythagorean triples is: $$a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2$$ where $m$ and $n$ are positive integers with $m > n$. 3. Important rules: - $m$ and $n$ must be positive integers. - $m$ must be greater than $n$. - This formula generates primitive triples (no common divisor) if $m$ and $n$ are coprime and not both odd. 4. Example: Let $m=3$ and $n=2$. Calculate $a$: $$a = 3^2 - 2^2 = 9 - 4 = 5$$ Calculate $b$: $$b = 2 \times 3 \times 2 = 12$$ Calculate $c$: $$c = 3^2 + 2^2 = 9 + 4 = 13$$ 5. Check the triple: $$5^2 + 12^2 = 25 + 144 = 169 = 13^2$$ which satisfies the Pythagorean theorem. 6. Therefore, $(5, 12, 13)$ is a Pythagorean triple.