Let's' Start with the Basics: What is “what”? 

The Pythagorean Theorem is like a movie star in the world of mathematics, everybody knows them. There is no standardized test without a question involving the triples. It’s one of the most widely studied theorems in the world. So, without any further delay, let’s get right to it. 

A Pythagorean Theorem is the sum of the squares of two shortest sides of a right-angled triangle is equal to the square of the longest side i.e. a hypotenuse.

So, to formulate, it comes to  a² + b² = c², where a and b are the two short sides and c is the hypotenuse or the longest side. So, a.b,c are whole numbers that make up the Pythagorean relationship. These sets of whole numbers are known as the Pythagorean Triples. So, these set of three whole numbers can be used for a right triangle, that establishes the Pythagorean relationship. 

Example: 

If the two short sides of the right-angled triangle are 3 cm and 4 cm, and the longest side or the hypotenuse is 5 cm, then:

(3)² + (4)² = (5)² 

         i.e. 

9 + 16 = 25 

What are primitive Pythagorean Triples? 

A primitive Pythagorean triples is where the length of two legs and the hypotenuse of a triangle are co-primes. Example: 3,4,5 is a set of primitive Pythagorean Triples. However, their multiples 6,8 and 10 respectively are not. 

If a,b and c are considered as the set of primitive Pythagorean triples, then here are some properties which you’d like to know

1. Either of a and b is odd while the other one will always be even 

2. c here will always be odd

3. Either of a and b will always be divisible by 3 

4. Either of a and b will always be divisible by 4

5. One from a,b and c will be divisible by 5

Finding more Pythagorean Triples

Now, if you feel like finding more triples, and to see if any set of whole numbers you have has a possibility of being a right-angled triangle, then proceed with the blog. 

Note: There can be more than two right-angled triangles that share the same hypotenuse. 

24² +7² = 25² = 15² + 20²

So, essentially when we are given a right-angled triangle with its respective lengths, it can be enlarged by doubling its lengths to get another right-angled triangle. That also means that we can produce as many triples we can with na, nb, nc where n is any whole number. So, the Pythagorean Triples set is endless. 

For instance, if you have a set of triples 3,4,5 and if we take n=5, then we get a set of new triples i.e. 15,20,25. The process works backward too. Let's take another set of triples 10,24,26. Now, if we halve them, we get 5,12,13 where each of them are whole numbers. But there is one exception to this case where reducing the lengths doesn't always procure whole numbers. For instance, 3,4,5 when halved doesn't give us a set of whole numbers. 

Euclid’s proof of infinite Pythagorean Triples

Apart from the explanation given above, Euclid had devised a different reasoning to prove that the set of Pythagorean Triples are endless. The foundation of Euclidean proof was based on the difference established between squares of two consecutive numbers which comes out as an odd number 

For example: 

2² − 1² = 4 − 1 = 3 

3² − 2² = 9 − 4 = 5

4² − 3² = 16 − 9 = 7

Here the differences seen are all odd values. 

Here’s a table for you understand this. 

So, you can see there are an infinite set of odd numbers because the perfect squares form a subset of odd numbers. And, a fraction of infinity is infinite as well. This proves that there should be an infinite number of odd squares and thus an infinite set of Pythagorean Triples. 

Properties of Pythagorean Triples

Here are a few interesting facts about Pythagorean Triples 

1. They always consists of all even numbers 

2. Alternatively, they can consist of an even number or two odd numbers

3. They can never constitute of two even numbers and odd numbers or all odd numbers  

• This above statement is true since the square of any odd number is also an odd number while the square of an even number is even 

• The sum of any two even numbers will be an even number while the sum of an odd number with that of an even number will give an odd number. 

It is quite easy to make sets of Pythagorean Triples, here’s a simple formula for that: 

Let’s take m and n as two positive integers where m > n 

• a = m² - n²

• b = 2mn

• c = m² + n²

Here, a,b and c represent the Pythagorean Triples

Example: 

Let’s test the formula out. We have m=2 and n=1 

Then, with the formula we can find the values of a,b and c

a = 2² − 1² = 3

b = 2 × 2 × 1 = 4

c = 2² + 1² = 5

What about the other methods to find the Pythagorean Triples? Does Odd Number has the same effect as the even ones while getting a set of triples? Are there any exceptions to it? 

Of course, these are the questions which you must be having. The topic just skims through the basics and there is more to it. As you can already see, we teach each of these topics with the sensitivity and patience required to help our students fully grasp the concept. Our interactive teaching session can be one of the best-implemented teaching methods that ensure longer retention. We help children see through these concepts without making them feel like a burden. 

Mathnasium of Olney aims to help students understand concepts like Pythagorean Triples so that they can excel through it without feeling the pressure. We hope to erase the fear and anxiety associated with mathematics and live our motto, which is we make math make sense!