# Pythagorean Triples

Last Updated : 15 Apr, 2024

Pythagorean Triples are used to explain the relationship between the three sides (i.e., a, b, and c) of the right triangle. These triples are a set of three positive integers, a, b, and c, that satisfy the equation aÂ² + bÂ² = cÂ².Â It is based on Pythagoras theorem which states that in any right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides of the triangle.Â Â

In this article, we will learn about the Pythagorean triples, its formulas, List of Pythagorean triples along with some examples.

## What are Pythagorean Triplets?

Pythagorean triples are use to find the three positive integers that satisfy the Pythagoras theorem. Generally, these three terms can be written in the form (a, b, c), and form a right-angle triangle with c as its hypotenuse and a and b as its base and height. The triangle formed by these terms is known as the Pythagorean triangle.

Let us consider a right-angled triangle in which b is the base, a is perpendicular, and c is the hypotenuse. So, according to the Pythagoras theorem: the sum of squares of sides a and b is equal to the square of the third side c.

a2 + b2 = c2

Here, a, b, and c are base, perpendicular, and hypotenuse of right angle triangle.

Now in this case we say that a, b, and c are Pythagorean Triples.

### Pythagorean Triples Examples

There are infinitely many possible Pythagorean triples as we can choose any two numbers for base and perpendicular and we can find hypotenuse using the Pythagoras theorem. For example, let’s say the perpendicular of the triangle is 4 units, and the base is 3 units, then the hypotenuse will be 5 using the Pythagoras theorem. This is further explained in the image added below.

Apart from these, many other Pythagorean triplets can be generated with the help of these basic Pythagorean triples(i.e. 3, 4, 5). The best way to obtain more triples is to scale them up, as all the integral multiple of any Pythagorean triplet is also a Pythagorean triple i.e., as (3, 4, 5) is Pythagorean triple which implies that (3n, 4n, and 5n) is always a Pythagorean triple, where, n âˆˆ {1, 2, 3, 4, 5, . . . }. Below is the illustration of the same:

n

(3n, 4n, 5n)

2

(6, 8, 10)

3

(9, 12, 15)

4

(12, 16, 20)

5

(15, 20, 25)

### Common Pythagorean Triples

The most commonly used Pythagorean Triples are (3, 4, 5). Other than this there are more examples of common Pythagorean triples such as (5, 12, 13), (6, 8, 10), (9, 12, 15), (7, 24, 25), and (15, 20, 25).

## Pythagorean Triples Formula

Pythagorean Triples Formula is derived from the right-angled triangle. The sides of the right-angle triangle arranged in increasing order as triples are Pythagorean triples. If two values out of three in a Pythagorean triple is given, the third can be obtained from the Pythagoras theorem which is also known as Pythagorean Triplets Formula.

The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the base and perpendicular. Let’s say the perpendicular is denoted by ‘a’, the base is denoted by ‘b’, and the hypotenuse is denoted by ‘c’, then the Pythagorean triples formula will be:

c2 = a2 + b2

## Proof of Pythagorean Triples Formula

Proof of the Pythagorean triplets Formula or Pythagoras theorem can be done in many ways. There are well over 371 proofs for this formula. Here we are using one of the many geometric methods. In this method, we use the figure as follows:

Step 1: We have four right-angled triangles with base m, perpendicular n, and hypotenuse p. Now arrange these triangles so that they make two squares one is outer square ABCD, whose side is m+n, and another one is inner square WZXY, whose side is p.

Step 2: Now we find the area of the inner square, outer square, and triangles:

Area of outer square, ABCD = (m + n)2

Area of inner square, WXYZ = (p)2

Area of one triangle = 1/2(m Ã— n)

â‡’ Area of four triangles (as all four triangles are the same) = 4 Ã— 1/2(m Ã— n)

â‡’ Area of four triangles = 2(m Ã— n)

Step 3: As we know that the area of square ABCD = Area of square WXYZ + Area of four triangles

â‡’ Â (m + n)2 = Â 2(m Ã— n) + p2Â

â‡’ m2 + 2 Ã— m Ã— n + n2 = 2 Ã— m Ã— n + p2

â‡’ m2 + n2 = p2

Hence ,Pythagorean triples Formula is proved.

## How to Form Pythagorean Triples?

Pythagorean triples are the positive integers and there are two cases for the number that can help us generate Pythagorean triples. The numbers can either be odd or even. The cases mentioned above can be explained in detail, as follows:

### If Number is Odd

If the generator number (m) is Odd then the following formula can be used to find the other two numbers to form a triple. If we have an odd number as a generator i.e., 1, 3, 5, 7, 9, . . ., then using m the remaining two numbers of triple can be found by putting m in the formula.

[m, (m2 – 1)/2 , (m2 + 1)/2]

Note: Here m should be greater than 1.

Example: If m = 3, find the rest of the Pythagorean triplets.

Solution:

We have m =3, which is an odd number

Hence, we will put the value of m in [m, (m2 – 1)/2 , (m2 + 1)/2]

(m2 -1)/2 = (32 – 1)/2 = 8/2 = 4.

and (m2 + 1)/2 = (32 + 1)/2 = 10/2 = 5

Therefore, the Pythagorean triples are (3, 4, 5).

### If Number is Even

If the generator number (m) is Even then we can use a different formula to find the other two numbers to form a triple. If we have an even number as a generator i.e., 2, 4, 6, 8,10, . . ., then using m the remaining two numbers of triple can be found by putting m in the Â following formula:

[m, (m2 – 4)/4, (m2 + 4)/4

Note: Here m should be greater than 2.

Example: Find the rest of the Pythagorean triples if m = 4.

Solution:

We have m = 4, which is an even number

Hence, we will put the value of [m, (m2 – 4)/4, (m2 + 4)/4]

(m2 – 4)/4 = (42 – 4)/4 = 12/4 = 3.

(m2 + 4)/4 = (42 + 4)/4 = 20/4 = 5.

Therefore, the Pythagorean triples are (3, 4, 5).

Note: Even if the method helps solve and find infinitely many Pythagorean triples, it still cannot find them all. For instance, the Pythagorean triples (20, 21, 29) cannot be formed using this technique. Thus, this formula is not absolute to find all possible Pythagorean triples.

## How to Generate Pythagorean Triples?

Other than the method illustrated above in this article, there is another way to generate Pythagorean Triples. In order to generate Pythagorean triples, we can assume the sides of the right-angled triangle are a, b, and c and define these sides in terms of two integral values m and n, such that,

• a is the perpendicular of the triangle here, and a = 2mn.
• b is the base of the triangle, and b = m2 – n2.
• c is the hypotenuse of the triangle, and c = m2 + n2.

Note: Here, m and n are the co-prime numbers such that m>n.

These values of a and b in terms of m and n, clearly satisfy Pythagoras Theorem, which is shown as follows:

c2 = a2 + b2

â‡’ (m2 + n2)2 = (2mn)2 + (m2 – n2)2

â‡’ m4 + n4 + 2m2n2 = 4m2n2 + m4 + n4 – 2m2n2

â‡’ m4 + n4 + 2m2n2 = m4 + n4 + 2m2n2

Thus, LHS = RHS,Â

Now simply use co-prime natural numbers in order to find the values of Pythagorean triplets. It is important to note that m must be greater than n.

Example: Find the Pythagorean triples when the values of m and n are 3 and 2, respectively.

Solution:

Pythagorean triples can be given as

• a = 2mn
• b = m2 – n2
• c = m2 + n2

Therefore, putting m = 3 and n = 2.

a = 2 Ã— 3 Ã— 2 = 12 units.

b = 32 – 22 = 9 – 4 = 5 units

c = 33 + 22 = 9 + 4 = 13 units.

Therefore, the Pythagorean triples are (5, 12, 13).

## Pythagorean Triples List

Below is the list of some of the Pythagorean triplets where the value of c (the hypotenuse of the triangle) is greater than 100:

 (20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125) (88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149) (85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181) (57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197) (84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221) (60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257) (23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)

These all values verify the Pythagorean triples formula a2 + b2 = c2.

## Types of Pythagorean Triples

Pythagorean Triples can further be classified into two types namely:

• Primitive Pythagorean Triples
• Non-Primitive Pythagorean Triples

### Primitive Pythagorean Triples

Primitive Pythagoras triples are also known as Reduced triples. The greatest common factor of these triples is 1. Or we can say that primitive Pythagorean triples are those triples in which the three numbers do not have any common divisor other than one. Such type of triples only contains one even number among the three given numbers.

Example: 3, 4, 5 is a primitive Pythagorean triple.

As 3, 4, 5 satisfy the Pythagorean triples formula and also the greatest common factor of (3, 4, 5) is 1.

### Non-Primitive Pythagorean TriplesÂ

Non-primitive Pythagoras triples are also known as imprimitive Pythagorean triples. Non-primitive Pythagorean triples are those triples in which the three numbers have a common divisor. Such types of triplets can contain more than one even positive number among the three given numbers.

Example: 6, 8, 10 is a Non-Primitive Pythagorean triple.

As, 6, 8, 10 satisfy the Pythagorean triples formula but the greatest common factor of 6, 8, 10 is 2 which is not equal to 1.

## Pythagorean Triples Properties

For a right-angled triangle with base m, height n, and hypotenuse p, Pythagorean triples have the following properties:

• All Pythagorean triples are positive integers.
• Pythagorean triples can be represented as m, n, p, or (m, n, p).
• Pythagorean triples always satisfy the formula m2 + n2 = p2.

### Fun Facts!

• There are infinite Pythagorean Triples.
• Pythagorean Triplet can either consist of all even numbers, or two odd numbers and one even number.
• All three numbers of a Pythagorean Triplet can never be odd.

## Triangular Numbers

We know that a triangular number is a number that can be arranged to form a triangle using the number of tiles as the number itself. As it is evident that the difference between two successive squares is successive odd numbers which suggests that every square is the sum of two successive triangular numbers.

And subsequently, the triangular numbers are the sums of all successive integers smaller than them. Such as,

• 0 + 1 = 1
• 0 + 1 + 2 = 3
• 0 + 1 + 2 + 3 = 6

Thus, some of the triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, etc.

## Pythagorean Triples Solved Examples

Example 1: Find Pythagorean triples if m = 8.

Solution:

This is the case when the number is even:

Given m = 8,Â

So, (m2 – 4)/4 = (64 – 4)/4 = 15

and (m2 + 4)/4 = (64 + 4)/4 = 17

Hence Pythagorean triplets are 8, 15, 17.

Example 2: Find Pythagorean triples if m = 9.

Solution:

This is the case when the number is odd:

Given m = 9,

So, (m2 – 1)/2 = (81 – 1)/2 = 40

and (m2 + 1)/2 = (81 + 1)/2 = 41

Hence Pythagorean triples are 9, 40, 41.

Example 3: Find Pythagorean triples, one of whose members is 13.

Solution:

Take m = 13,

So, (m2 – 1)/2 = (169 – 1)/2 = 84

and (m2 + 1)/2 =(169 + 1)/2 = 85

Hence Pythagorean triples are 13, 84, 85.

Example 4: Check if (6, 8, 10) is a Pythagorean triplet or not.

Solution:

Let us take m = 6, n = 8, and p = 10

According to the formulaÂ

m2 + n2 = p2

â‡’ (6)2 + (8)2 = (10)2

â‡’ 36 + 64 = 100

â‡’ 100 = 100

Here, L.H.S = R.H.S

Hence, (6, 8, 10) is a Pythagorean triple.

Example 5: If (y, 84, 85) is a Pythagorean triplet, then find the value of y.

Solution:

Let us take m = y, n = 84, and p = 85

According to the formulaÂ

m2 + n2 = p2

â‡’ (y)2 + (84)2 = (85)2

â‡’ y2 + 7056 = 7225

â‡’ y2 = 169

â‡’ y = 13 [as y can’t be negative]

## Practice Question on Pythagorean Triples

Question 1: Find the Pythogorean triples of 21.

Question 2: Prove that (12, 35, 37) is a Pythagorean triple.

Question 3: Find x, if (11, x, 61) is a Pythagorean triple.

Question 4: Find the other two numbers of a Pythagorean triple, if one of the number is 5.

Question 5: Check if (4, 7, 9) is a Pythagorean triple or not.

## Summary – Pythagorean Triples

The article discusses Pythagorean triples, which are sets of three integers (a, b, c) that satisfy the equation a2 + b2 = c2 and represent the sides of a right triangle. It explains how these triples can be generated using formulas based on whether the initial integer (m) is odd or even, offering specific examples such as (3, 4, 5) and (5, 12, 13). The article also covers methods for creating these triples by manipulating two co-prime integers, m and n, ensuring that m is greater than n, to satisfy the Pythagorean theorem. This exploration of Pythagorean triples illustrates their fundamental role in geometric and algebraic concepts, emphasizing their practical and theoretical significance in mathematics.

## Pythagorean Triples FAQs

### What are the Pythagorean Triples?

Pythagorean triples are the natural numbers that satisfy the Pythagoras theorem. Therefore, the Pythagorean triples satisfy the formula c2 = a2 + b2, here, c is the hypotenuse of the right-angled triangle, a and b are the legs of the triangle.

### What are the five most common Pythagorean Triplets?

The five most common Pythagorean triples are:Â

• (3, 4, 5)
• (5, 12, 13)
• (6, 8, 10)
• (9, 12, 15)
• (15, 20, 25)

### How to Find Pythagorean triples?

The Pythagorean triples can be easily obtained by the help of pythagoras theorem, if two triples are already given, the third triplet can be found:

a2 + b2 = c2

Where, a and b are the legs of the triangle and c is the hypotenuse of the triangle.

### Can Pythagorean Triples be Decimals?

No, Pythagorean triples can never be decimal as they are positive integers, that is, natural numbers and satisfy the Pythagoras theorem.

### What is the Scaling of Triples?

The scaling of the pythagoras triples is multiplying the given triples with some natural numbers and observing that the new set generated satisfies the Pythagorean triples condition too. For example, a Pythagorean triplet (3, 4, 5) is given to us, if we multiply the triples with 3, it will give (9, 12, 15).

### What is Pythagoras Triples of 8?

The Pythagoras Triplets of 8 are 8, 6, and 10 as,

82 + 62 = 102

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