## The mysterious number 495

There are certain numbers that have certain characteristics.   Until now, many of these numbers become source of fascinations of mathematicians and non-mathematicians alike. Many mathematicians have spent their entire lives solving the mysteries of some numbers.  In this post, we are going to talk about the mysterious number 495.
Let’s have  a game.
1. Think of a 3-digit number where not all the digits are the same.
2. Arrange the digits in descending order and ascending order from left to right. This will give you two 3-digit numbers — one the smallest number that you an form from the digits, and the other the largest.
3. Subtract the smaller number from the larger number.
4. Go back to step 2.
If you repeat this process over and over again, you will end up with 495! » Read more

## 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