### Theory:

Triplet means a set of three numbers.
A triplet $$(a, b, c)$$ of three natural numbers is called a Pythagorean triplets if it satisfies ${a}^{2}+{b}^{2}={c}^{2}$.
Example:
1. Check $$(3, 4, 5)$$ is a Pythagorean triplet.

Solution:

Here $$a = 3$$, $$b = 4$$ and $$c = 5$$.

L.H.S. $$=$$ $$a^2 + b^2$$

$$= 3^2 + 4^2$$

$$= 9 + 16$$

$$= 25$$

R.H.S. $$=$$ $$c^2$$

$$= 5^2 = 25$$

So, $$25$$ $$=$$ $$25$$

L.H.S. $$=$$ R.H.S.

Therefore, $$(3, 4, 5)$$ is a Pythagorean triplet.

2. Check $$(6, 8, 9)$$ is a Pythagorean triplet or not.

Solution:

Here $$a = 6$$, $$b = 8$$ and $$c = 9$$.

L.H.S. $$=$$ $$a^2 + b^2$$

$$= 6^2 + 8^2$$

$$= 36 + 64$$

$$= 100$$

R.H.S. $$=$$ $$c^2$$

$$=$$ $$9^2 = 81$$

So, $$100$$ $$\ne$$ $$81$$

L.H.S. $$\ne$$ R.H.S.

Therefore, $$(6, 8, 9)$$ is not a Pythagorean triplet.
General form of Pythagorean triplet
Let us consider any natural $$a > 1$$.

The triplet $$(2a, a^2 - 1, a^2 + 1)$$ will form a Pythagorean triplet.

The general formula to find a Pythagorean triplet is $$(2a)^2 + (a^2 - 1)^2 = (a^2 + 1)^2$$, for any natural number $$a > 1$$.
Example:
Find a Pythagorean triplet one of whose least number is $$10$$.

General form of Pythagorean triplet is $$(2a, a^2 - 1, a^2 + 1)$$.

Least number $$(2a) = 10$$

$$a = \frac{10}{2}$$

$$a = 5$$

$$a^2 - 1 = 5^2 - 1 = 24$$

$$a^2 + 1 = 5^2 + 1 = 26$$

The triplet is $$(10, 24, 26)$$.

To check the triplet satisfies the Pythagorean relation:

$$(2a)^2 + (a^2 - 1)^2 = (a^2 + 1)^2$$

$$10^2 + 24^2 = 26^2$$

$$100 + 576 = 676$$

$$676 = 676$$

Therefore, $$(10, 24, 26)$$ is a Pythagorean triplet.
Important!
L.H.S. $$=$$ Left Hand Side

R.H.S. $$=$$ Right Hand Side