Important Problems on Pythagorean Triples

This section features a collection of important problems on Pythagorean Triples, designed to help students apply their understanding through practice. Each problem highlights key concepts such as generating triples, identifying right-angled triangles, and using properties of integers. The problems range from basic to advanced levels and include step-by-step solutions for clarity. Ideal for school and college students, math Olympiad participants, and exam aspirants, this guide enhances your problem-solving skills with a strong focus on Pythagorean Triples in both geometry and number theory.

Verify that \(3,4,5\) is the only primitive Pythagorean triple involving consecutive positive integers.

Answer:

We are looking for positive integers \(a, b, c\) such that:

a^2 + b^2 = c^2,\quad \text{and} \quad a \lt b \lt c,\quad \text{with} \quad a, b, c \quad \text{consecutive}.

Let us assume:

a = n, \quad b = n+1, \quad c = n+2

Substitute into the Pythagorean identity:

\begin{align*} & a^2 + b^2 = c^2 \\ \Rightarrow \quad & n^2 + (n+1)^2 = (n+2)^2 \\ \Rightarrow \quad & n^2 + n^2 + 2n + 1 = n^2 + 4n + 4 \\ \Rightarrow \quad & 2n^2 + 2n + 1 = n^2 + 4n + 4 \\ \Rightarrow \quad & 2n^2 + 2n + 1 - (n^2 + 4n + 4) = 0 \\ \Rightarrow \quad & n^2 - 2n - 3 = 0 \\ \Rightarrow \quad & (n-3)(n+1) = 0 \\ \Rightarrow \quad & n = 3 \quad \text{or} \quad n = -1 \end{align*}

Since we are interested in positive integers, taking \(n = 3\). So:

a = 3,\quad b = 4,\quad c = 5

Since:

\gcd(3, 4, 5) = 1

Hence, the only primitive Pythagorean triple involving consecutive positive integers is \(3, 4, 5\).