# HOW TO FIND PYTHAGOREAN TRIPLET FROM ONE NUMBER

Consider the following,

32 + 42  =   52

The collection of numbers 3, 4 and 5 is known as Pythagorean triplet.

Relationship between Pythagorean Triplet :

Square of larger number

=  Sum of squares of other two small numbers

If the given numbers will have the above relationship, we can say the given numbers are pythagorean triplets.

For any natural number m > 1, we have

(2m)2 +  (m2 - 1)2  + (m2 + 1)2

So, 2m, (m2 - 1) and (m2 + 1) forms a Pythagorean triplet.

Example 1 :

Find the Pythagorean triplet in which one number is 8.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).

Let us consider the given number as 2m

2m  =  8

m  =  4

 (m2 - 1)  =  (42 - 1)     =  16 - 1     =  15 (m2 + 1)  =  (42 + 1)=  16 + 1 =  17

The triplet is 8, 15 and 17.

Verifying the relationship :

172  =  152 + 82

289  =  225 + 64

289  =  289

Example 2 :

Find the Pythagorean triplet in which one number is 12.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).

Let us consider the given number as 2m

2m  =  12

m  =  6

 (m2 - 1)  =  (62 - 1)=  36 - 1 =  35 (m2 + 1)  =  (62 + 1)=  36 + 1 =  37

The triplet is 12, 35 and 37.

Verifying the relationship :

372  =  352 + 122

1369  =  1225 + 144

1369  =  1369

Example 3 :

Find the Pythagorean triplet in which one number is 14.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).

Let us consider the given number as 2m

2m  =  14

m  =  7

 (m2 - 1)  =  (72 - 1)=  49 - 1 =  48 (m2 + 1)  =  (72 + 1)=  49 + 1 =  50

The triplet is 14, 48 and 50.

Verifying the relationship :

502  =  482 + 142

2500  =  2304 + 196

2500  =  2500

Example 4 :

Find the Pythagorean triplet in which one number is 6.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).

Let us consider the given number as 2m

2m  =  6

m  =  3

 (m2 - 1)  =  (32 - 1)=  9 - 1 =  8 (m2 + 1)  =  (32 + 1)=  9 + 1   =  10

The triplet is 6, 8 and 10.

Verifying the relationship :

102  =  82 + 62

100  =  64 + 36

100  =  100

Example 5 :

Find the Pythagorean triplet in which one number is 16.

Solution :

We can get the Pythagorean triplet by using the general form 2m, (m2 - 1), (m2 + 1).

Let us consider the given number as 2m

2m  =  16

m  =  8

 (m2 - 1)  =  (82 - 1)=  64 - 1 =  63 (m2 + 1)  =  (82 + 1)=  64 + 1 =  65

The triplet is 16, 63 and 65.

Verifying the relationship :

652  =  632 + 162

4225  =  3969 + 256

4225  =  4225

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