Pythagorean Triples

Let’s get started with the basics by knowing what are Pythagorean triplets.

  • According to Pythagoras theorem, the square of the length of the hypotenuse of a right triangle is the sum of the squares of the lengths of the other two sides. 

  • The Pythagoras theorem can be usually expressed as p2+q2 = r2

 

Here’s a formula for Pythagoras theorem,

          p2+q2 = r2

 

Now, when we replace the variables p, q, r with some integers and if they satisfy the above equation then they are known as Pythagorean triples. 

 

The most common examples of Pythagorean triples are (3, 4 , 5) and (5,12,13). 

 

Now, you might think that the Pythagorean Theorem can be used for any triangle.

But the theorem is applicable only for a right-angle triangle.

What Is A Right-Angle Triangle?

To understand what a right angle triangle is, let us consider a right-angle triangle named ABC, with its three sides namely the opposite, adjacent, and the hypotenuse. In a right-angled triangle, we generally refer to the three sides according to their relation with the angle. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.            

The three sides of a right-angled triangle are as follows-

  • The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse (H). 

  • The side that is opposite to the angle θ is known as the opposite (O). 

  • The side which lies next to the angle   is known as the Adjacent(A)

     Pythagoras theorem states that, 

In any right-angled triangle, 


(Opposite)2+(Adjacent)2= (Hypotenuse)2


Something you need to know!

When we multiply the entries in a triple by any integer and get another Pythagorean triple. 

For example :- (6 , 8,10), (9,12,15) and (15,20,25).

How to Find Pythagorean Triples?

Here are the rules of how to find Pythagorean Triples,

  • Each and every odd number is the p side of a Pythagorean triplet( p2+q2 = r2)

  • The q side in a Pythagorean triplet is equally to (p2 – 1)/2.

  • The r side is equal to q + 1.

Now, we know that p and r are always odd and q is even.

  • These relationships are true because the difference between successive square numbers is the successive odd numbers.

  • All the odd numbers in the relationship are itself a square (and the square of all odd numbers is an odd number itself) which gives us a Pythagorean triplet.

Primitive Pythagorean Triples:

The triples for which the entries are relatively prime are known as Primitive Pythagorean Triples. But all three legs in the Primitive Pythagorean triples cannot be prime.

As we can see in the common example (3, 4, 5), the numbers 3 and 5 are prime numbers and here 4 is an even number.

So in any given primitive Pythagorean triple one of the three entries must be even.

 

The most elementary number theory texts prove that all primitive triples (p,q, r) are given by the following:

           

  p = u2 – v2, q= 2uv, r = u2 + v2

 

In the above equation u and v are relatively prime integers, not both odd. We observe that, p is a difference of squares, so for it to be prime we need that u and v differ by 1. So, we can write,

             

  a = 2v + 1, b = 2v2 + 2v, and c = 2v2 + 2v + 1.

 

Here are the first few triples we then expect to see infinitely many triples with two primes according to Schinzel and Sierpinski's Hypothesis H.

The Pythagorean triplets list consists of all the possible Pythagorean triplets.

 

Pythagorean Triplets list:

Prime leg

Even leg

Hypotenuse

3

4

5

5

12

13

11

60

61

19

180

181


These are some Pythagorean triples examples. 

Questions to be Solved :

Question 1) Which of the following is not a Pythagorean triple?

a) 7, 24, 25

b) 8, 15, 17

c) 9, 12, 15

d)10,16,19

Answer) Pythagorean Triples are sets of whole numbers which fit the rule:
a2 + b2 = c2

In Option a, 72 + 242 = 49 + 576 = 625 = 252 which means 7, 24 and 25 is a Pythagorean triple.
In Option b, 82 + 152 = 64 + 225 = 289 = 172 which means 8, 15, and 17 is a Pythagorean triple.
In Option c, 92 + 122 = 81 + 144 = 225 = 152 which means 9, 12 and 15 is a Pythagorean triple.
In Option d, 102 + 162 = 100 + 256 = 356 ≠ 192 which means 10, 16 and 19 is not a Pythagorean triple.

(7, 24, 25 ),( 8, 15, 17), (9, 12, 15) are Pythagorean triples examples.

Question 2) Check if (8, 15,17) are Pythagorean triples?

Solution: Given, Pythagorean triples = (8, 15, 17)

We can say that p = 17, q = 15, r =8

We know the Pythagorean triples formula is, p2 = q2 + r2

LHS, p= 172 = 289

RHS, r2 + q2 = 82 + 152 

= 64 +225 = 289

We see that in the given question,

LHS = RHS

Therefore, the given number set (7, 24, 25) is one of the Pythagorean triple examples.

FAQ (Frequently Asked Questions)

1. What are examples of Pythagorean triples and what is the smallest Pythagorean triples?

The Pythagorean theorem has been derived from the Pythagorean triples proof which states that integer triples which satisfy this equation are known as Pythagorean triples. The most common examples are (3,4,5) and (5,12,13) that are very common in Mathematics. Notice that when we multiply the entries in a triple by any integer, we get another Pythagorean triple. For example, (6, 8,10), (9,12,15) and (15,20,25).The smallest Pythagorean Triple in Mathematics is 3, 4 and 5 in Mathematics.