## Guest Post: Vedic Mathematics 4 – Nikhilam Method of Multiplication

Sanjay Guilati (author) has been teaching computer and mathematics in Bhilai, state Chattisgarh in India for 15 years. Currently he is a  teacher in a senior secondary school and he is also involved in teacher training.  His online work can be found in Mathematics Academy.

Multiplication – Nikhilam (निखिलम) Method

Conventional method of multiplication is time consuming. It involves number of steps and space. Multiplications done using Vedic Mathematics is easy, requires less number of steps, requires less time. One question can be solved in many ways and can be done mentally.

Nikhilam multiplication is one such method in which problems of special patterns can be solved in a step or two. For using this method, one must be familiar with the base system, which is described below. » Read more

## Pythagorean Triple 2: Generating Pythagorean Triples

In the previous math article, we have shown that there are infinitely many Pythagorean triples.  In this article, we are going to discuss a very short but effective strategy in generating Pythagorean Triples.

A Pythagorean triple is the integer triple $(a,b, c)$ satisfying the Pythagorean equation $a^2 + b^2 = c^2$.

Observe the Pythagorean triples $(3, 4, 5)$ and  $(5, 12, 13)$.  We can see that the hypotenuse is greater than the longer side by $1$. From the pattern, we can form the Pythagorean triples $(a, b, b + 1)$ satisfying the equation $a^2 + b^2 = (b + 1)^2$.

Right triangle with side length 3, 4 and 5 units.

Solving the equation we have $a^2 = 2b + 1$, which implies that  $a = \sqrt{2b +1}$.  Now,   $2b + 1$ is always odd (can you see why?). It follows that in order for $a$ to be an integer, $2b + 1$ must be a perfect square. This means, that we are sure that $a$ is an integer, if $2b + 1$ is an odd perfect square.

From here, we can generate infinitely many examples of Pythagorean triples.  For example, $49$ is an odd perfect square. So plugging it in the equation we have,  $2b + 1 = 49$, then, we have the triple $(7, 24, 25)$, another Pythagorean Triple.  If we let $2b + 1 = 121$, then we have the triple $(11, 60, 61)$.

Now, we found another way to generate Pythagorean Triples.

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## Prime Series 1: Introduction to Prime Numbers

We have learned from elementary school mathematics that a prime number has only two factors, 1 and itself. For example, 2, 3, 5 and 7 are prime numbers, while 8 is not prime because it has four factors — 1, 2, 4, and 8. Numbers that are not prime are called composite numbers.

Geometric Interpretation of Prime and Composite Numbers

Mathematicians during the ancient times, particularly the Greeks, always make  use of geometric interpretations of numbers. Square numbers, for example, are represented with pebbles arranged with the same number of rows and columns.  The first five square numbers are 1, 4, 9, 16 and 25.

Rectangular numbers are also popular. For instance, the number 12 can be interpreted pebbles arranged in rectangles with the following dimensions: 3 by 4, 2 by 6 and 12 by 1. If we are going to use squares instead of pebbles, the geometric representations of these arrangements are shown in Figure 1.

Figure 1 – Different rectangular arrangements of 12 pebbles represented by squares.

Numbers that cannot be arranged as more than one rectangle are prime numbers. In our example above, 12 has three possible arrangements, while the numbers 3, 5 and 7 can only be arranged in a single row.

Figure 2 – Rectangular arrangements of 3, 5 and 7 pebbles represented by squares.

As we can see, this is the geometric interpretation of the definition that the factors of primes are only 1 and itself.

Sieve of Eratosthenes

The Greek philosopher and mathematician Eratosthenes was the first to be credited in identifying primes in a finite list by brute force.  The strategy is to list a finite set of counting numbers in increasing order, then starting with 2 eliminate all its multiples. This eliminates all even numbers except 2. Then we follow the same pattern: we eliminate multiples of 3 greater than 3, eliminate all multiples of 5 greater than 5 and so on, until all the numbers left are not multiples of any number smaller than it. The remaining numbers after all elimination are prime numbers.

The primes numbers less than 100 are shown in white cells in the table below. The numbers in the yellow cells are composite numbers.  Mathematicians agreed not to include 1 in the set of prime numbers or the set of composite numbers.

Figure 3 - The sieve of Eratosthenes shows primes less than 100 in white table cells..

If we look at the table above, we could not probably see a pattern about the number of primes in a given interval; however, if we investigate further, as the intervals increase, the number of primes is getting fewer and fewer. For example, there are 168 primes between 1 and 1000, 135 primes between 1000 and 2000, 127 primes between 2000 and 3000, and 120 primes between 3000 and 4000. With this observation, we want to ask the following question:

Is there a particular prime number, that after such number, we could no longer find primes?

or equivalently,

Are prime numbers finite?

We will answer this in the continuation of this article titled  “Infinitude of Primes.” The formal proof of this conjecture is also discussed in the third part.

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