A Pythagorean Triplet has 3 numbers a, b, c
and
a ^{ 2 } + b ^{ 2 } = c ^{ 2 }
Thus, we say that (a, b, c) are Pythagorean triplet
Note : This a, b, c are sides of a right triangle
The most common Pythagorean Triplets are
 3, 4, 5
 5, 12, 13
 7, 24, 25

8, 15, 17
Let’s see how we check them
For Numbers 3, 4, 5
3 ^{ 2 } + 4 ^{ 2 } = 9+16
= 25
= 5 ^{ 2 }
∴ 3
^{
2
}
+4
^{
2
}
=5
^{
2
}
Thus, 3, 4, 5 are Pythagoras Triplets.
For Number 5, 12, 13
5 ^{ 2 } + 1222 = 25 + 144
= 169
= 13 ^{ 2 }
Thus, 5, 12, 13 are Pythagorean Triplets.
In General
(2m) ^{ 2 } + (m ^{ 2 }  1)2 = (m ^{ 2 } + 1) ^{ 2 }
Here,
First number = 2m
Second number = 2m
Third number = m ^{ 2 } +1
from Pythagoras Triplets.
Note:
If we know that 3, 4, 5 are Pythagorean Triplets
Then,
3 × 2 = 6
4 × 2 = 8
5 × 2 = 10
So, (6, 8, 10) will also be a Pythagorean Triplet
Similarly, if we multiply by 3
(9, 12, 15) is also a Pythagorean Triplet
And, more generally, if we multiply by any number k
(3k, 4k, 5k) will also be a Pythagorean Triplet