## Generating Pythagorean Triples from Square Numbers

A figurate number is a number that can be represented by a regular geometrical arrangement of equally spaced points (or circles as shown in the first figure). If the arrangement forms a regular polygon, the number is called a polygonal number.

Examples of polygonal numbers are square numbers. The first  four square numbers are 1, 4, 9, and 16, and their geometric representations are shown in the first figure. It is clear that that the 10th square number has 102 circles, and in general, the nth square number has n2 circles.

Looking at the color pattern above, we can see that there is something very special about square numbers. Each square number can be represented as the sum of odd integers.  The first four examples are shown below. » Read more

## The Infinitude of Pythagorean Triples

In the Understanding the Fermat’s Last Theorem post, I have mentioned about Pythagorean Triples.  In this post, we will show that there are infinitely many of them. We will use intuitive reasoning to prove the theorem.

For the 100th time (kidding), recall that the Pythagorean Theorem states that in a right triangle with side lengths $a, b$ and $c$, where $c$ is the hypotenuse, the equation $c^2 = a^2 + b^2$  is satisfied. For example, if we have a triangle with side lengths $2$ and $3$ units, then the hypotenuse is $\sqrt{13}$. The converse of the Pythagorean theorem is also true: If you have side lengths, $a, b$ and $c$, which satisfies the equation above, we are sure that the angle opposite to the longest side is a right angle.

We are familiar with right triangles with integral sides. The triangle with sides $(3, 4, 5)$ units, for instance, is a right triangle.  This is also the same with $(5, 12, 13)$ and $(8, 15, 17)$.  We will call this triples, the Pythagorean triples ,or geometrically, right triangles having integral side lengths. » Read more

## The Algebraic and Geometric Proofs of Pythagorean Theorem

The Pythagorean Theorem states that if a right triangle has side lengths $a, b$ and $c$, where $c$ is the hypotenuse, then the sum of the squares of the two shorter lengths is equal to the square of the length of the hypotenuse.

Figure 1 – A right triangle with side lengths a, b and c.

Putting it in equation form, we have

$a^2 + b^2 = c^2$.

For example, if a right triangle has side lengths $5$ and $12$, then the length of its hypotenuse is $13$, since $c^2 = 5^2 + 12^2 \Rightarrow c = 13$.

Exercise 1: What is the hypotenuse of the triangle with sides $1$ and $\sqrt{3}$?

The converse of the theorem is also true. If the side lengths of the triangle satisfy the equation $a^2 + b^2 = c^2$, then the triangle is right. For instance, a triangle with side lengths $(3, 4, 5)$ satisfies the equation $3^2 + 4^2 = 5^2$, therefore, it is a right triangle.

Geometrically, the Pythagorean theorem states that in a right triangle with sides $a, b$ and $c$ where $c$ is the hypotenuse, if three squares are constructed whose one of the sides are the sides of the triangle as shown in Figure 2, then the area of the two smaller squares when added equals the area of the largest square.

Figure 2 – The geometric interpretation of the Pythagorean theorem states that the area of the green square plus the area of the red square is equal to the area of the blue square.

One specific case is shown in Figure 3: the areas of the two smaller squares are $9$ and $16$ square units, and the area of the largest square is $25$ square units.

Exercise 2: Verify that the area of the largest square in Figure 3 is 25 square units by using the unit squares.

Figure 3 – A right triangle with side lengths 3, 4 and 5.

Similarly, triangles with side lengths $(7, 24, 25)$ and  $(8, 15, 17)$ are right triangles. If the side lengths of a right triangle are all integers, we call them Pythagorean triples. Hence, $(7, 24, 25)$ and  $(8, 15, 17)$ are Pythagorean triples.

Exercise 3: Give other examples of Pythagorean triples.

Exercise 4: Prove that there are infinitely many Pythagorean triples.

Proofs of the Pythagorean Theorem

There are more than 300 proofs of the Pythagorean theorem. More than 70 proofs are shown in tje Cut-The-Knot website. Shown below are two of the proofs.  Note that in proving the Pythagorean theorem, we want to show that for any right triangle with hypotenuse $c$, and sides $a$, and $b$, the following relationship holds: $a^2 + b^2 = c^2$.

Geometric Proof

First, we draw a triangle with side lengths $a, b$ and $c$ as shown in Figure 1. Next, we create 4 triangles identical to it and using the triangles form a square with side lengths $a + b$ as shown in Figure 4-A. Notice that the area of the white square in Figure 4-A is $c^2$.

Figure 4 – The Geometric proof of the Pythagorean theorem.

Rearranging the triangles, we can also form another square with the same side length as shown in Figure 4-B.This means that the area of the white square in the Figure 4-A is equal to the sum of the areas of the white squares in Figure 4-B (Why?). That is, $c^2 = a^2 + b^2$ which is exactly what we want to show. *And since we can always form a (big) square using four right triangles with any dimension (in higher mathematics, we say that we can choose arbitrary $a$ and $b$ as side lengths of a right triangle), this implies that the equation $a^2 + b^2 = c^2$ stated above is always true regardless of the size of the triangle.

Exercise 5: Prove that the quadrilateral with side length C in Figure 4-A is a square.

Algebraic Proof

In the second proof, we will now look at the yellow triangles instead of the squares.  Consider Figure 4-A. We can compute the area of a square with side lengths $a + b$ using two methods: (1) we can square the side lengths and (2) we can add the area of the 4 congruent triangles and then add them to the area of the white square which is $c^2$.  If we let $A$ be the area of the square with side $b + a$, then calculating we have

Method 1: $A = (b + a)^2 = b^2 + 2ab +a^2$

Method 2:  $A = 4(1/2ab) + c^2 = 2ab + c^2$

Methods 1 and 2 calculated the area of the same square, therefore they must be equal. This means that we can equate both expressions.  Equating we have,

$b^2 + 2ab + a^2 = 2ab + c^2 \Rightarrow a^2 + b^2 = c^2$

which is exactly what we want to show.